Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 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)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,40)(29,35)(30,36)(31,37)(32,38)(33,39)(41,48)(42,46)(43,45)(44,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,37,46)(26,40,39,35)(32,41,44,43)(38,48,47,45);(10,14)(11,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,37)(26,38)(28,41)(29,43)(30,42)(33,44)(35,45)(36,46)(39,47)(40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(19,28)(21,33)(23,35)(24,37)(27,41)(29,30)(32,44)(34,45)(36,40)(42,43)(46,48);(2,5)(6,8)(10,14)(13,15)(20,31)(24,37)(27,42)(32,44)(34,46)(38,47)(41,43)(45,48)Polycyclic presentation:
2^9|b^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7455776);