Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #42664 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1457 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)(10,14)(11,12)(17,18,22,27)(19,24,30,39)(20,32,35,25)(21,26,31,40)(23,28,34,42)(29,45,47,38)(33,44,37,41)(36,48,46,43);(9,10)(11,16)(12,15)(13,14)(17,19,22,30)(18,24,27,39)(20,33,35,37)(21,34,31,23)(25,41,32,44)(26,42,40,28)(29,46,47,36)(38,48,45,43);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20,34,46,22,35,23,36)(18,25,42,48,27,32,28,43)(19,29,21,37,30,47,31,33)(24,38,26,44,39,45,40,41);(9,12,13,11)(10,15,14,16)(20,35)(25,32)(29,47)(33,37)(36,46)(38,45)(41,44)(43,48);(1,4,5,3)(2,6,7,8)(17,21,23,19,22,31,34,30)(18,26,28,24,27,40,42,39)(20,33,36,47,35,37,46,29)(25,41,43,45,32,44,48,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744696);