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)(33,34,38,39)(35,43,46,60)(36,41,47,54)(37,51,52,42)(40,55,53,45)(44,61,48,64)(49,59,50,58)(56,62,57,63);(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)(36,47)(40,53)(42,51)(43,60)(44,48)(49,50)(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)(33,35,37,45,38,46,52,55)(34,40,42,43,39,53,51,60)(36,48,50,62,47,44,49,63)(41,57,59,64,54,56,58,61);(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)(35,44)(37,50)(38,47)(39,54)(40,56)(42,59)(43,61)(45,63)(46,48)(49,52)(51,58)(53,57)(55,62)(60,64);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,26)(22,30)(25,27)(29,31)(33,37,38,52)(34,42,39,51)(35,45,46,55)(36,49,47,50)(40,43,53,60)(41,58,54,59)(44,62,48,63)(56,64,57,61)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=fgi,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, 7180158);