Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1685 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,32)(21,26)(22,38)(23,28)(24,31)(25,34)(27,36)(30,41)(33,43)(35,42)(37,40)(39,48)(44,45)(46,47);(9,10)(11,15)(12,14)(13,16)(17,19,22,31,23,21,36,37)(18,24,27,29,28,40,38,26)(20,30,35,44,34,33,47,48)(25,39,42,43,32,45,46,41);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,28,32)(19,30,21,33)(22,35,36,47)(24,39,40,45)(26,41,29,43)(27,42,38,46)(31,44,37,48);(11,16)(13,15)(17,21,23,19)(18,26,28,29)(20,33,34,30)(22,31,36,37)(24,27,40,38)(25,41,32,43)(35,44,47,48)(39,42,45,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,34)(25,32)(30,33)(35,47)(39,45)(41,43)(42,46)(44,48)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6997662);