Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| 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 | #42685 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1222 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)(33,34,38,40)(35,45,48,56)(36,42,50,54)(37,44,39,43)(41,47,53,49)(46,59,60,64)(51,58,52,57)(55,62,63,61);(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)(34,40)(36,50)(41,53)(43,44)(45,56)(46,60)(51,52)(61,62);(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)(33,35,37,47,38,48,39,49)(34,41,43,56,40,53,44,45)(36,46,52,62,50,60,51,61)(42,55,58,64,54,63,57,59);(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)(34,42)(35,46)(37,52)(38,50)(39,51)(40,54)(41,55)(43,58)(44,57)(45,59)(47,62)(48,60)(49,61)(53,63)(56,64);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,26)(22,30)(25,27)(29,31)(33,37,38,39)(34,43,40,44)(35,47,48,49)(36,51,50,52)(41,56,53,45)(42,57,54,58)(46,61,60,62)(55,59,63,64)Polycyclic presentation:
2^9|a^2=fh,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=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7180090);