Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 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)(11,16)(12,14)(13,15)(17,18,23,29)(19,25,32,41)(20,33,37,26)(21,27,24,30)(22,28,39,46)(31,47,48,40)(34,44,36,42)(35,45,38,43);(1,2,4,5)(3,6,7,8)(17,19)(18,25)(20,34)(21,22)(23,32)(24,39)(26,42)(27,28)(29,41)(30,46)(31,38)(33,44)(35,48)(36,37)(40,45)(43,47);(9,11,13,12)(10,14,15,16)(17,20,21,35,23,37,24,38)(18,26,27,43,29,33,30,45)(19,31,39,36,32,48,22,34)(25,40,46,44,41,47,28,42);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,27,29,30)(19,22,32,39)(20,35,37,38)(25,28,41,46)(26,43,33,45)(31,34,48,36)(40,42,47,44);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,22,21,19,23,39,24,32)(18,28,27,25,29,46,30,41)(20,36,35,48,37,34,38,31)(26,44,43,47,33,42,45,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744195);