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 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1392 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)(17,18,24,30)(19,25,32,39)(20,26,36,43)(21,27,23,29)(22,28,37,44)(31,38,45,47)(33,40,46,48)(34,41,35,42);(1,2)(3,6)(4,5)(7,8)(17,19,24,32)(18,25,30,39)(20,33,36,46)(21,22,23,37)(26,40,43,48)(27,28,29,44)(31,34,45,35)(38,41,47,42);(9,11,13,12)(10,14,15,16)(17,20,23,35,24,36,21,34)(18,26,29,42,30,43,27,41)(19,31,22,33,32,45,37,46)(25,38,28,40,39,47,44,48);(1,3,4,7)(2,6,5,8)(17,21,24,23)(18,27,30,29)(19,22,32,37)(20,34,36,35)(25,28,39,44)(26,41,43,42)(31,33,45,46)(38,40,47,48);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,22,23,32,24,37,21,19)(18,28,29,39,30,44,27,25)(20,33,35,45,36,46,34,31)(26,40,42,47,43,48,41,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743595);