Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510385 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31555 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,17)(4,24)(5,30)(6,14)(7,13)(8,15)(9,16)(10,20)(11,27)(12,34)(18,64)(19,52)(21,59)(22,46)(23,57)(25,79)(26,77)(28,78)(29,50)(31,44)(32,53)(33,61)(35,51)(36,58)(37,55)(38,56)(39,98)(40,49)(41,54)(42,62)(43,68)(45,72)(47,82)(48,84)(60,89)(63,67)(65,101)(66,102)(69,104)(70,114)(71,90)(73,108)(74,118)(75,106)(76,81)(80,107)(83,127)(85,91)(86,119)(87,110)(88,116)(92,128)(93,103)(94,115)(95,111)(96,100)(97,117)(99,112)(105,124)(109,125)(113,121)(120,123)(122,126);(1,3)(2,10,16,46)(4,25,29,82)(5,31,34,69)(6,20)(7,22)(8,39)(9,23)(11,47)(12,52)(13,17,14,57)(15,59,62,106)(18,28,68,89)(19,32,71,37)(21,36)(24,76)(26,83)(27,81,35,73)(30,90)(33,70,41,74)(38,75)(40,67,96,66)(42,72)(43,100)(44,55)(45,56,98,58)(48,109,91,127)(49,64)(50,108)(51,79)(53,104)(54,105)(60,102)(61,103)(63,78)(65,99)(77,126,110,107)(80,85)(84,122)(86,121)(87,125)(88,92)(93,97,124,94)(95,123)(101,119,120,116)(111,128,112,113)(114,117)(115,118);(1,4,9,29)(2,11)(3,18)(5,26,34,87)(6,35,7,27)(8,40,42,96)(10,43,46,64)(12,48)(13,50)(14,24)(15,60)(16,51)(17,63,57,102)(19,65)(20,67)(21,73)(22,66)(23,68)(25,39)(28,38,89,36)(30,91)(31,92)(32,85,37,84)(33,95,41,99)(44,101,104,120)(45,76,98,108)(47,106,79,59)(49,58)(52,113,90,128)(53,110)(54,116)(55,77)(56,100)(61,119)(62,78)(69,121)(70,83)(71,123)(72,82)(74,125)(75,81)(80,93)(86,97,88,94)(103,127,105,109)(107,114,126,118)(111,117)(112,115)(122,124);(1,5)(2,12,16,30)(3,19,23,71)(4,26)(6,32)(7,37)(8,41)(9,34)(10,44)(11,48,51,91)(13,53,14,55)(15,61,62,54)(17,52)(18,65,68,123)(20,31,22,69)(21,74,75,70)(24,77,50,110)(25,80,82,122)(27,84)(28,88)(29,87)(33,42)(35,85)(36,97)(38,94)(39,93,72,124)(40,99)(43,101)(45,105)(46,104)(47,107)(49,112,100,111)(56,117,58,115)(57,90)(59,118)(60,119,78,116)(63,113)(64,120)(66,121,67,92)(73,125,81,83)(76,109)(79,126)(86,89)(95,96)(98,103)(102,128)(106,114)(108,127)Polycyclic presentation:
2^9|b^2=fh,c^2=eh,d^2=ehi,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417582);