Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,28)(20,32)(21,26)(22,25)(23,30)(24,36)(27,34)(29,43)(31,39)(33,46)(35,41)(37,47)(38,44)(40,45)(42,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,33)(21,31)(22,30)(24,40)(25,28)(26,38)(27,39)(29,35)(32,43)(34,44)(36,47)(37,42)(41,46)(45,48);(2,6)(5,7)(10,13)(14,16)(17,20,34,48,22,35,21,36)(18,24,26,41,25,42,27,32)(19,29,44,47,30,33,31,45)(23,37,38,43,28,40,39,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,35)(26,27)(31,44)(36,48)(37,40)(43,46);(3,8)(5,7)(10,13)(14,16)(17,21,22,34)(18,26,25,27)(19,31,30,44)(20,36,35,48)(23,38,28,39)(24,41,42,32)(29,45,33,47)(37,43,40,46)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6691987);