Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1495 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)(11,18)(12,26)(13,19)(14,32)(15,35)(16,23)(17,28)(20,34)(21,31)(22,48)(24,54)(25,39)(27,60)(29,63)(30,59)(33,47)(36,46)(37,66)(38,68)(40,50)(41,61)(42,64)(43,56)(44,53)(45,62)(49,58)(51,69)(52,65)(55,67)(57,71)(70,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,37)(18,19)(20,38)(21,39)(26,58)(27,49)(28,50)(29,51)(30,52)(31,53)(32,65)(33,55)(34,56)(35,44)(36,57)(40,41)(42,67)(43,45)(46,69)(47,70)(48,60)(54,59)(61,66)(62,68)(63,71)(64,72);(9,12,13,27)(10,17,19,41)(11,22,23,49)(14,33,30,64)(15,36,31,29)(16,37,18,40)(20,46,45,63)(21,47,35,42)(24,55,52,72)(25,57,53,51)(26,59,60,32)(28,62,61,34)(38,69,43,71)(39,70,44,67)(48,65,58,54)(50,68,66,56);(3,8)(5,7)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,19)(17,40)(20,43)(21,44)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,39)(36,57)(37,41)(38,45)(42,70)(46,71)(47,67)(58,60)(59,65)(61,66)(62,68)(63,69)(64,72);(2,6)(5,7)(10,19)(11,23)(17,41)(20,45)(21,35)(22,49)(24,52)(25,53)(26,60)(32,59)(42,47)(46,63)(50,66)(51,57)(55,72)(56,68)Polycyclic presentation:
2^9|c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531418);