Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | |
1657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,29)(20,24)(21,26)(22,39)(23,28)(25,32)(27,30)(31,44)(33,40)(34,41)(35,42)(36,43)(37,45)(38,48)(46,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,30,34,23,32,22,20)(18,24,39,25,28,41,27,29)(21,35,47,33,37,31,38,36)(26,43,48,44,45,40,46,42);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,32,23,34,30,19)(18,25,27,24,28,29,39,41)(21,36,38,31,37,33,47,35)(26,44,46,43,45,42,48,40);(11,16)(13,15)(17,21)(18,26)(19,31)(20,33)(22,38)(23,37)(24,40)(25,42)(27,46)(28,45)(29,44)(30,47)(32,35)(34,36)(39,48)(41,43);(10,14)(13,15)(21,37)(26,45)(31,35)(33,36)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413770);