Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | |
1969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,31)(21,35)(22,26)(23,27)(25,34)(28,45)(29,41)(30,39)(32,48)(33,43)(36,40)(37,44)(38,47)(42,46);(1,2,4,5)(3,8,7,6)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,24,26,39)(20,32,34,46)(21,29,23,36)(25,42,31,48)(27,40,35,41)(28,37,47,33)(38,43,45,44);(10,14)(13,16)(17,20)(18,25)(19,28)(21,33)(22,34)(23,37)(24,38)(26,31)(27,43)(29,46)(30,47)(32,36)(35,44)(39,45)(40,42)(41,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,26)(24,39)(25,31)(27,35)(38,45)(40,41)(42,48)(43,44);(17,21,22,23)(18,27,26,35)(19,29,30,36)(20,33,34,37)(24,40,39,41)(25,43,31,44)(28,46,47,32)(38,42,45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7413625);