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)(17,18,23,29)(19,30,34,25)(20,27,21,26)(22,38,24,28)(31,40,35,44)(32,41,48,47)(33,42,39,43)(36,45,37,46);(2,6)(7,8)(11,16)(13,15)(17,19,23,34)(18,25,29,30)(20,35,21,31)(22,33,24,39)(26,44,27,40)(28,42,38,43)(32,36,48,37)(41,46,47,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,23)(22,37)(24,36)(25,40)(27,29)(28,46)(30,44)(32,39)(33,48)(34,35)(38,45)(41,42)(43,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,23,20)(18,27,29,26)(19,32,34,48)(22,37,24,36)(25,41,30,47)(28,46,38,45)(31,33,35,39)(40,43,44,42);(2,6)(7,8)(17,22,23,24)(18,28,29,38)(19,33,34,39)(20,36,21,37)(25,42,30,43)(26,45,27,46)(31,32,35,48)(40,47,44,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=f,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, 7440224);