Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #24249 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,25)(21,35)(22,27)(23,26)(24,31)(29,45)(30,39)(32,48)(33,44)(34,43)(36,40)(37,42)(38,47)(41,46);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,27,28)(20,32,34,46)(21,30,23,36)(25,41,43,48)(26,39,35,40)(29,37,47,33)(38,42,45,44);(10,13)(14,16)(17,20,22,34)(18,25,27,43)(19,29,31,47)(21,33,23,37)(24,38,28,45)(26,42,35,44)(30,46,36,32)(39,41,40,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,42,43,44)(29,46,47,32)(38,41,45,48);(3,8)(5,7)(19,30)(21,23)(24,40)(26,35)(28,39)(29,46)(31,36)(32,47)(33,37)(38,48)(41,45)(42,44)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6602526);