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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,34)(21,38)(22,27)(23,26)(24,32)(25,37)(29,40)(30,41)(31,42)(33,46)(35,43)(36,44)(39,45)(47,48);(2,6)(7,8)(11,16)(13,15)(17,19,22,32)(18,24,27,28)(20,35,37,29)(21,33,23,31)(25,43,34,40)(26,46,38,42)(30,39,48,36)(41,44,47,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,39)(22,37)(23,36)(24,40)(26,45)(27,34)(28,43)(30,33)(31,48)(32,35)(38,44)(41,42)(46,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,39)(24,41)(26,44)(27,34)(28,47)(29,33)(31,35)(32,48)(38,45)(40,42)(43,46);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,38)(19,31,32,33)(20,36,37,39)(24,42,28,46)(25,44,34,45)(29,30,35,48)(40,47,43,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7442979);