Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497148 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #22276 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,16,21,52)(4,23)(5,29,33,51)(6,12)(7,36)(8,15)(9,19,41,22)(10,27)(11,31,50,84)(13,34)(14,35)(17,55,59,108)(18,61)(20,66,56,40)(24,72,60,92)(25,75,78,119)(26,69)(28,71)(30,48)(32,49,88,98)(37,53)(38,74,93,114)(39,64)(42,95,97,58)(43,89)(44,99,101,128)(45,79)(46,103)(47,81)(54,65,106,67)(57,110)(62,105)(63,117)(68,76,118,111)(70,80)(73,120)(77,82,122,102)(83,123,90,87)(85,107)(86,115)(91,125)(94,116)(96,124)(100,112)(104,109)(113,126)(121,127);(1,3,6,22,8,21,35,19)(2,9,12,52,15,41,14,16)(4,24,26,17,28,60,80,59)(5,30,84,18,33,62,31,64)(7,20,34,67,37,56,89,65)(10,42,45,38,47,97,46,93)(11,48,51,61,50,105,29,39)(13,40,53,106,43,66,36,54)(23,55,69,92,71,108,70,72)(25,57,111,73,78,113,76,85)(27,74,79,58,81,114,103,95)(32,86,123,63,88,116,87,109)(44,100,102,127,101,91,82,96)(49,104,90,115,98,117,83,94)(68,110,119,120,118,126,75,107)(77,112,128,121,122,125,99,124);(1,4,7,27,35,80,89,103,8,28,37,81,6,26,34,79)(2,10,43,71,12,45,36,70,15,47,13,23,14,46,53,69)(3,17,56,95,19,24,67,114,21,59,20,58,22,60,65,74)(5,25,32,77,84,111,123,128,33,78,88,122,31,76,87,99)(9,38,40,92,52,97,106,108,41,93,66,72,16,42,54,55)(11,44,98,118,29,82,90,119,50,101,49,68,51,102,83,75)(18,57,109,125,62,73,86,124,64,113,63,112,30,85,116,121)(39,91,94,126,105,127,117,120,61,100,115,110,48,96,104,107);(1,5,8,33)(2,11)(3,18)(4,25,28,78)(6,31,35,84)(7,32,37,88)(9,39,41,61)(10,44)(12,29)(13,49)(14,51)(15,50)(16,48,52,105)(17,57)(19,62)(20,63)(21,64)(22,30)(23,68)(24,73)(26,76,80,111)(27,77,81,122)(34,87,89,123)(36,90)(38,91,93,100)(40,94,66,115)(42,96,97,127)(43,98)(45,82)(46,102)(47,101)(53,83)(54,104,106,117)(55,107,108,120)(56,109)(58,112)(59,113)(60,85)(65,116)(67,86)(69,75)(70,119)(71,118)(72,110,92,126)(74,121)(79,99,103,128)(95,125)(114,124)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efhi,d^2=eh,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,g^b=gi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404345);