Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18,22,27)(19,23,30,26)(20,31,33,36)(21,25,35,42)(24,38,40,28)(29,37,46,43)(32,47,34,48)(39,45,41,44);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,19,22,30)(18,23,27,26)(20,28,33,38)(21,29,35,46)(24,36,40,31)(25,37,42,43)(32,44,34,45)(39,48,41,47);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,20,30,38,22,33,19,28)(18,24,23,36,27,40,26,31)(21,34,46,44,35,32,29,45)(25,41,37,47,42,39,43,48);(1,3,5,4)(2,8,6,7)(9,12,14,13)(10,15,11,16)(17,21,22,35)(18,25,27,42)(19,29,30,46)(20,32,33,34)(23,37,26,43)(24,39,40,41)(28,44,38,45)(31,47,36,48);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,15,11,16)(21,35)(25,42)(29,46)(32,34)(37,43)(39,41)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6959349);