Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 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)(3,6)(4,5)(7,8)(17,18,22,27)(19,28,30,24)(20,32,33,42)(21,26,23,37)(25,34,41,35)(29,44,46,48)(31,40,36,39)(38,45,43,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,40)(27,28)(32,48)(33,46)(34,47)(35,45)(37,39)(41,43)(42,44);(10,13)(14,16)(17,20,22,33)(18,25,27,41)(19,29,30,46)(21,35,23,34)(24,38,28,43)(26,42,37,32)(31,47,36,45)(39,48,40,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,33)(24,28)(25,41)(31,36)(32,42)(34,35)(39,40);(11,15)(14,16)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,34,33,35)(24,39,28,40)(25,32,41,42)(29,45,46,47)(38,44,43,48)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6441318);