Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,32)(15,36)(16,21)(18,29)(20,34)(22,47)(23,39)(24,40)(25,53)(26,56)(28,58)(30,59)(31,46)(33,42)(35,61)(37,45)(38,50)(41,55)(43,62)(44,60)(48,68)(49,69)(51,71)(52,66)(54,64)(57,67)(63,70)(65,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,24)(14,25)(15,26)(16,23)(18,38)(19,40)(20,41)(21,39)(27,47)(28,49)(29,50)(30,48)(31,52)(32,53)(33,54)(34,55)(35,51)(36,56)(37,57)(42,64)(43,65)(44,63)(45,67)(46,66)(58,69)(59,68)(60,70)(61,71)(62,72);(3,8)(5,7)(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,42)(20,45)(21,44)(23,48)(24,49)(25,54)(26,57)(27,32)(29,34)(31,60)(35,62)(36,43)(39,63)(40,64)(41,67)(46,58)(47,53)(50,55)(51,72)(52,70)(56,65)(59,61)(66,69)(68,71);(2,6)(5,7)(11,23)(17,39)(22,48)(24,26)(25,51)(38,63)(40,56)(41,66)(47,68)(49,57)(50,70)(52,55)(53,71)(54,72)(64,65)(67,69);(9,13,16,15)(10,19,21,36)(11,24,23,26)(12,28,30,37)(14,31,35,34)(17,40,39,56)(18,42,44,43)(20,32,46,61)(22,49,48,57)(25,52,51,55)(27,58,59,45)(29,33,60,62)(38,64,63,65)(41,53,66,71)(47,69,68,67)(50,54,70,72)Polycyclic presentation:
2^9|e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530500);