Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(6,8)(9,10)(11,16)(12,25)(13,30)(14,35)(15,18)(17,27)(19,33)(20,32)(21,47)(22,38)(23,52)(24,55)(26,57)(28,63)(29,59)(31,46)(34,58)(36,45)(37,48)(39,54)(40,51)(41,60)(42,62)(43,65)(44,61)(49,70)(50,71)(53,69)(56,68)(64,66)(67,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,41)(18,38)(19,39)(20,40)(21,29)(25,57)(27,60)(28,56)(30,52)(31,64)(32,51)(33,54)(34,50)(35,55)(36,49)(37,42)(43,69)(44,67)(45,70)(46,66)(47,59)(48,62)(53,65)(58,71)(61,72)(63,68);(9,12,15,29)(10,17,18,42)(11,21,22,26)(13,31,34,65)(14,36,32,28)(16,37,38,41)(19,45,44,63)(20,46,35,43)(23,53,50,64)(24,56,51,49)(25,58,59,30)(27,61,62,33)(39,68,67,70)(40,69,55,66)(47,71,57,52)(48,72,60,54);(2,5)(6,8)(10,18)(16,38)(17,42)(19,44)(20,35)(25,59)(30,58)(37,41)(39,67)(40,55)(43,46)(45,63)(47,57)(52,71)(66,69)(68,70);(1,3,4,7)(2,6,5,8)(10,18)(11,22)(17,42)(19,44)(20,35)(21,26)(23,50)(24,51)(25,59)(30,58)(43,46)(45,63)(48,60)(49,56)(53,64)(54,72)Polycyclic presentation:
2^9|c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,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, 7531260);