Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,37,22,28,21,27)(19,24,30,40,32,42,31,41)(20,26,34,25,36,44,35,45)(29,39,48,38,33,46,47,43);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,30)(25,43)(26,46)(27,41)(28,42)(29,36)(34,47)(35,48)(37,40)(38,45)(39,44);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,44)(28,45)(30,47)(31,48)(32,33)(39,40)(41,46)(42,43);(2,6)(5,7)(17,20)(18,26)(19,29)(21,35)(22,36)(23,34)(24,39)(25,37)(27,45)(28,44)(30,48)(31,47)(32,33)(38,40)(41,43)(42,46);(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,30,32,31)(20,34,36,35)(24,40,42,41)(25,44,45,26)(29,47,33,48)(38,39,43,46)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6665333);