Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| 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 |
| prime radical | #61086 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10,15,18)(11,16,22,40)(12,25,29,61)(13,30,34,60)(14,35,32,20)(17,41,43,27)(19,46,45,33)(21,49,26,59)(23,54,52,72)(24,57,53,39)(28,63,36,47)(31,48,65,44)(37,62,42,50)(38,67,69,56)(51,71,58,68)(55,70,64,66);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,42)(18,40)(19,38)(20,39)(21,29)(25,59)(27,62)(28,58)(30,54)(31,64)(32,53)(33,56)(34,52)(35,57)(36,51)(37,43)(41,50)(44,70)(45,69)(46,67)(47,71)(48,66)(49,61)(55,65)(60,72)(63,68);(9,12,15,29)(10,17,18,43)(11,21,22,26)(13,31,34,65)(14,36,32,28)(16,37,40,42)(19,47,45,63)(20,48,35,44)(23,55,52,64)(24,58,53,51)(25,60,61,30)(27,46,41,33)(38,68,69,71)(39,70,57,66)(49,72,59,54)(50,67,62,56);(3,8)(5,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,37)(18,40)(19,38)(20,39)(25,49)(26,29)(27,50)(28,51)(30,54)(31,55)(32,53)(33,56)(34,52)(35,57)(36,58)(41,62)(42,43)(44,66)(45,69)(46,67)(47,68)(48,70)(59,61)(60,72)(63,71)(64,65);(10,18)(11,22)(17,43)(19,45)(20,35)(21,26)(23,52)(24,53)(25,61)(30,60)(44,48)(47,63)(50,62)(51,58)(55,64)(56,67)Polycyclic presentation:
2^9|a^2=fi,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530976);