Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #35757 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2255 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,34,43)(20,26,33,42)(21,27,23,29)(22,28,31,40)(32,41,39,48)(35,44,38,47)(36,45,37,46);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,35)(21,32)(22,36)(23,39)(24,34)(26,44)(27,41)(28,45)(29,48)(30,43)(31,37)(33,38)(40,46)(42,47);(10,14)(13,16)(17,20,24,33)(18,26,30,42)(19,31,34,22)(21,37,23,36)(25,40,43,28)(27,46,29,45)(32,38,39,35)(41,47,48,44);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,32,34,39)(20,36,33,37)(22,38,31,35)(25,41,43,48)(26,45,42,46)(28,47,40,44);(3,8)(5,7)(17,22,24,31)(18,28,30,40)(19,33,34,20)(21,38,23,35)(25,42,43,26)(27,47,29,44)(32,37,39,36)(41,46,48,45)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585303);