Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #24435 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,12,13)(11,15,16,14)(17,18,22,27)(19,23,29,37)(20,32,34,40)(21,25,35,26)(24,39,41,33)(28,43,44,48)(30,38,31,42)(36,45,47,46);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,31)(22,29)(24,36)(25,42)(26,38)(27,37)(30,35)(32,43)(33,46)(34,44)(39,45)(40,48)(41,47);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,20,21,33,22,34,35,39)(18,24,26,32,27,41,25,40)(19,28,31,46,29,44,30,45)(23,36,38,43,37,47,42,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,29)(23,37)(28,44)(30,31)(36,47)(38,42)(43,48)(45,46);(2,6)(5,7)(17,21,22,35)(18,25,27,26)(19,30,29,31)(20,33,34,39)(23,38,37,42)(24,40,41,32)(28,45,44,46)(36,43,47,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818208);