Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18)(19,23)(20,31)(21,25)(22,27)(24,34)(26,33)(28,43)(29,37)(30,38)(32,46)(35,41)(36,45)(39,44)(40,47)(42,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,32)(21,30)(22,29)(24,40)(25,38)(26,39)(27,37)(28,35)(31,46)(33,44)(34,47)(36,42)(41,43)(45,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,33,48,22,35,21,34)(18,24,25,41,27,42,26,31)(19,28,44,47,29,32,30,45)(23,36,38,46,37,40,39,43);(9,11)(10,14)(12,15)(13,16)(19,29)(23,37)(28,32)(30,44)(36,40)(38,39)(43,46)(45,47);(11,15)(14,16)(17,21,22,33)(18,25,27,26)(19,30,29,44)(20,34,35,48)(23,38,37,39)(24,41,42,31)(28,45,32,47)(36,46,40,43)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,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, 6467336);