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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1501 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,24,30)(19,25,34,42)(20,26,21,27)(22,28,23,29)(31,39,36,44)(32,40,47,48)(33,41,38,46)(35,43,37,45);(2,6)(7,8)(11,16)(13,15)(17,19,24,34)(18,25,30,42)(20,31,21,36)(22,33,23,38)(26,39,27,44)(28,41,29,46)(32,35,47,37)(40,43,48,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,37)(23,35)(25,39)(27,30)(28,45)(29,43)(32,33)(34,36)(38,47)(40,41)(42,44)(46,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,24,20)(18,27,30,26)(19,32,34,47)(22,37,23,35)(25,40,42,48)(28,45,29,43)(31,38,36,33)(39,46,44,41);(2,6)(7,8)(17,22,24,23)(18,28,30,29)(19,33,34,38)(20,35,21,37)(25,41,42,46)(26,43,27,45)(31,47,36,32)(39,48,44,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7440226);