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 | #25109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1490 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)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,34,43)(20,26,37,46)(21,27,23,29)(22,28,35,44)(31,40,38,47)(32,41,39,48)(33,42,36,45);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,24,34)(18,25,30,43)(20,35,37,22)(21,32,23,39)(26,44,46,28)(27,41,29,48)(31,33,38,36)(40,42,47,45);(10,13)(14,16)(17,20,21,33,24,37,23,36)(18,26,27,42,30,46,29,45)(19,31,39,22,34,38,32,35)(25,40,48,28,43,47,41,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,24,23)(18,27,30,29)(19,32,34,39)(20,36,37,33)(22,38,35,31)(25,41,43,48)(26,45,46,42)(28,47,44,40);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,22,23,31,24,35,21,38)(18,28,29,40,30,44,27,47)(19,33,32,20,34,36,39,37)(25,42,41,26,43,45,48,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=fgi,e^2=gh,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, 6747796);