Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,10,13,7)(4,8,5,14)(11,16,12,15)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31)(33,34)(35,43)(36,41)(37,52)(38,39)(40,46)(42,44)(45,60)(47,59)(48,54)(49,64)(50,58)(51,57)(53,61)(55,63)(56,62);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29)(34,39)(36,48)(40,53)(42,52)(43,59)(45,49)(50,51)(62,63);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,35,44,61,38,47,37,46)(34,40,52,59,39,53,42,43)(36,49,50,63,48,45,51,62)(41,56,57,60,54,55,58,64);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(20,28)(23,32)(25,27)(29,31)(33,36)(34,41)(35,45)(37,51)(38,48)(39,54)(40,55)(42,58)(43,60)(44,50)(46,63)(47,49)(52,57)(53,56)(59,64)(61,62);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,26)(22,30)(25,27)(29,31)(33,37,38,44)(34,42,39,52)(35,46,47,61)(36,50,48,51)(40,43,53,59)(41,57,54,58)(45,62,49,63)(55,64,56,60)Polycyclic presentation:
2^9|a^2=fhi,b^2=fhi,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7180170);