Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | #25131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,26)(19,28,30,23)(20,32,35,42)(21,25,36,27)(24,40,43,34)(29,44,33,47)(31,39,46,38)(37,45,41,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,26,28)(20,33,35,29)(21,31,36,46)(24,41,43,37)(25,38,27,39)(32,44,42,47)(34,48,40,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,34,22,35,36,40)(18,24,27,32,26,43,25,42)(19,29,31,45,30,33,46,48)(23,37,39,47,28,41,38,44);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(23,28)(29,33)(31,46)(37,41)(38,39)(44,47)(45,48);(11,15)(14,16)(17,21,22,36)(18,25,26,27)(19,31,30,46)(20,34,35,40)(23,38,28,39)(24,42,43,32)(29,45,33,48)(37,44,41,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,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^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6467562);