Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 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)(9,10)(11,14)(12,15)(13,16)(17,18,22,23)(19,24,30,29)(20,31,34,25)(21,26,35,36)(27,32,41,37)(28,38,42,43)(33,44,46,39)(40,45,48,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15)(17,19,22,30)(18,24,23,29)(20,32,34,37)(21,28,35,42)(25,27,31,41)(26,38,36,43)(33,45,46,47)(39,40,44,48);(9,11,13,12)(10,15,16,14)(17,20,22,34)(18,25,23,31)(19,27,30,41)(21,33,35,46)(24,37,29,32)(26,39,36,44)(28,40,42,48)(38,47,43,45);(2,6)(5,7)(17,21)(18,26)(19,28)(20,33)(22,35)(23,36)(24,38)(25,39)(27,40)(29,43)(30,42)(31,44)(32,45)(34,46)(37,47)(41,48);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(18,23)(19,29)(21,35)(24,30)(25,31)(27,32)(28,38)(33,46)(37,41)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6631953);