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 | #25761 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 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)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,30,40)(20,31,33,24)(21,25,35,43)(27,37,29,39)(28,45,47,38)(32,48,46,41)(34,44,36,42);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,32)(21,27)(22,30)(24,41)(25,37)(26,40)(28,34)(29,35)(31,48)(33,46)(36,47)(38,42)(39,43)(44,45);(10,13)(14,16)(19,27)(20,33)(21,35)(23,37)(24,31)(25,43)(28,46)(29,30)(32,47)(38,48)(39,40)(41,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(20,33)(24,31)(28,47)(32,46)(34,36)(38,45)(41,48)(42,44);(3,8)(5,7)(17,20,22,33)(18,24,26,31)(19,28,30,47)(21,36,35,34)(23,38,40,45)(25,44,43,42)(27,32,29,46)(37,41,39,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6603217);