Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1481 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,34)(21,29)(22,27)(23,28)(24,33)(25,31)(26,35)(32,46)(36,43)(37,45)(38,41)(39,44)(40,48)(42,47);(2,5)(3,7)(17,19)(18,24)(20,35)(21,31)(22,38)(23,33)(25,26)(27,43)(28,30)(29,34)(32,47)(36,39)(37,48)(40,42)(41,44)(45,46);(9,11,12,16)(10,13,14,15)(17,20,35,19,23,31,21,33)(18,25,26,24,28,34,29,30)(22,36,47,32,39,38,37,48)(27,41,42,40,44,43,45,46);(2,5)(6,8)(17,21,23,35)(18,26,28,29)(19,31,33,20)(22,37,39,47)(24,25,30,34)(27,42,44,45)(32,38,48,36)(40,41,46,43);(1,3,4,7)(2,6,5,8)(17,22,21,37,23,39,35,47)(18,27,29,45,28,44,26,42)(19,32,20,36,33,48,31,38)(24,40,25,41,30,46,34,43)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6725366);