Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
1433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,23,30,39)(20,24,32,41)(21,25,35,44)(27,36,45,47)(28,37,29,38)(31,40,46,48)(33,42,34,43);(1,2,5,6)(3,7,4,8)(17,19)(18,23)(20,31)(21,28)(22,30)(24,40)(25,37)(26,39)(27,33)(29,35)(32,46)(34,45)(36,42)(38,44)(41,48)(43,47);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,22,32)(18,24,26,41)(19,27,30,45)(21,34,35,33)(23,36,39,47)(25,43,44,42)(28,31,29,46)(37,40,38,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(20,32)(24,41)(27,45)(31,46)(33,34)(36,47)(40,48)(42,43);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(19,28)(20,32)(21,35)(23,37)(24,41)(25,44)(27,46)(29,30)(31,45)(36,48)(38,39)(40,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6647732);