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 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10,14,20)(11,23,26,17)(12,28,30,62)(13,22,16,19)(15,35,34,46)(18,42,44,31)(21,33,47,37)(24,52,54,61)(25,40,32,45)(27,59,58,66)(29,63,38,48)(36,49,65,43)(39,55,50,70)(41,57,51,64)(53,71,56,69)(60,68,72,67);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,32)(14,26)(15,27)(16,25)(18,39)(19,45)(20,23)(21,41)(22,40)(28,61)(29,56)(30,54)(31,55)(33,64)(34,58)(35,66)(36,60)(37,57)(38,53)(42,70)(43,68)(44,50)(46,59)(47,51)(48,69)(49,67)(52,62)(63,71)(65,72);(9,12)(10,18)(11,24)(13,29)(14,30)(15,36)(16,38)(17,39)(19,43)(20,44)(21,48)(22,49)(23,50)(25,53)(26,54)(27,60)(28,35)(31,37)(32,56)(33,42)(34,65)(40,67)(41,69)(45,68)(46,62)(47,63)(51,71)(52,59)(55,57)(58,72)(61,66)(64,70);(2,5)(6,8)(9,13,14,16)(10,19,20,22)(11,25,26,32)(12,29,30,38)(15,33,34,37)(17,40,23,45)(18,43,44,49)(21,46,47,35)(24,53,54,56)(27,57,58,64)(28,48,62,63)(31,36,42,65)(39,67,50,68)(41,66,51,59)(52,69,61,71)(55,72,70,60);(1,3,4,7)(2,6,5,8)(10,20)(11,26)(18,44)(19,22)(21,47)(24,54)(25,32)(27,58)(28,62)(35,46)(43,49)(48,63)(53,56)(55,70)(57,64)(60,72)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531803);