Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509608 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30846 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13312 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 134 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)(6,37)(7,15)(8,14)(9,16)(10,20)(11,28)(12,51)(13,39)(18,63)(19,68)(21,74)(22,57)(23,62)(25,78)(26,81)(27,88)(29,49)(31,96)(32,58)(33,55)(34,54)(35,94)(36,95)(38,60)(40,53)(41,50)(42,103)(43,72)(44,69)(45,73)(46,65)(47,84)(48,91)(52,98)(56,101)(59,102)(61,66)(64,107)(67,116)(70,92)(71,106)(75,80)(76,120)(77,90)(79,119)(82,108)(83,109)(85,93)(86,122)(87,123)(89,105)(97,113)(99,115)(100,114)(104,118)(110,112)(111,126)(117,127)(121,124)(125,128);(1,3)(2,10)(4,25)(5,31)(6,38)(7,21)(8,22)(9,23)(11,46)(12,52)(13,57)(14,44)(15,60)(16,45)(17,37)(18,28)(19,33)(20,39)(24,75)(26,82)(27,89)(29,66)(30,92)(32,73)(34,98)(35,99)(36,100)(40,69)(41,67)(42,49)(43,54)(47,108)(48,61)(50,105)(51,70)(53,62)(55,96)(56,115)(58,74)(59,114)(63,77)(64,112)(65,91)(68,94)(71,101)(72,102)(76,121)(78,88)(79,84)(80,90)(81,107)(83,103)(85,124)(86,97)(87,125)(93,119)(95,106)(104,122)(109,116)(110,113)(111,128)(117,120)(118,126)(123,127);(1,4)(2,11)(3,18)(5,26)(6,27)(7,41)(8,28)(9,29)(10,42)(12,47)(13,48)(14,49)(15,24)(16,50)(17,61)(19,64)(20,65)(21,25)(22,66)(23,67)(30,93)(31,97)(32,83)(33,84)(34,86)(35,85)(36,87)(37,77)(38,80)(39,90)(40,91)(43,104)(44,105)(45,75)(46,60)(51,112)(52,113)(53,109)(54,81)(55,76)(56,111)(57,116)(58,88)(59,110)(62,78)(63,74)(68,119)(69,103)(70,118)(71,79)(72,117)(73,89)(82,99)(92,107)(94,123)(95,122)(96,108)(98,125)(100,124)(101,120)(102,126)(106,127)(114,128)(115,121);(1,5,32,101,8,34,6,33,9,36,39,51,7,35,40,102)(2,12,53,95,15,55,13,54,16,56,37,30,14,59,58,94)(3,19,69,98,21,71,20,31,23,72,38,99,22,70,73,100)(4,26,83,120,28,86,27,84,29,87,90,112,41,85,91,126)(10,43,74,96,44,106,17,52,45,68,57,114,60,92,62,115)(11,47,109,122,24,76,48,81,50,111,77,93,49,110,88,123)(18,64,103,125,25,79,65,97,67,117,80,82,66,118,89,124)(42,104,63,108,105,127,61,113,75,119,116,128,46,107,78,121)Polycyclic presentation:
2^9|d^2=ehi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416805);