Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1462 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,26)(20,31)(21,35)(22,27)(23,30)(24,28)(25,37)(29,38)(32,41)(33,46)(34,39)(36,42)(40,45)(43,48)(44,47);(1,3,7,4)(2,5,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,29,37,36)(24,39,41,31)(25,38,35,42)(33,45,47,43)(40,46,48,44);(1,4,7,3)(2,5,8,6)(10,13)(14,15)(17,20,19,28,22,34,30,32)(18,24,23,31,27,41,26,39)(21,33,29,43,37,47,36,45)(25,40,38,44,35,48,42,46);(1,5,7,6)(2,3,8,4)(9,11,12,16)(10,14,13,15)(17,21)(18,25)(19,29)(20,33)(22,37)(23,38)(24,40)(26,42)(27,35)(28,43)(30,36)(31,44)(32,45)(34,47)(39,46)(41,48);(1,6,7,5)(2,3,8,4)(17,19,22,30)(18,26,27,23)(20,28,34,32)(21,36,37,29)(24,39,41,31)(25,38,35,42)(33,45,47,43)(40,44,48,46)Polycyclic presentation:
2^9|a^2=hi,b^2=fhi,c^2=fgi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893126);