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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,30)(20,33)(21,36)(22,38)(23,29)(24,27)(25,32)(26,35)(28,40)(31,42)(34,43)(37,44)(39,45)(41,48)(46,47);(1,3)(2,5)(4,8)(6,7)(17,19)(18,25)(20,21)(22,39)(23,32)(24,35)(26,36)(27,33)(28,45)(29,30)(31,37)(34,40)(38,43)(41,47)(42,48)(44,46);(9,11,13,12)(10,14,15,16)(17,20,21,19,23,35,24,32)(18,26,36,25,29,33,27,30)(22,39,37,47,40,34,41,31)(28,45,48,46,38,43,44,42);(2,6)(5,7)(17,21,23,24)(18,27,29,36)(19,20,32,35)(22,37,40,41)(25,33,30,26)(28,44,38,48)(31,34,47,39)(42,45,46,43);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,22,21,37,23,40,24,41)(18,28,36,48,29,38,27,44)(19,31,35,39,32,47,20,34)(25,42,33,45,30,46,26,43)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6741811);