Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #38035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1537 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)(3,7)(4,6)(5,8)(17,18,22,26)(19,28,30,24)(20,32,34,47)(21,27,23,37)(25,33,41,35)(29,43,44,46)(31,40,36,39)(38,45,42,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,29,34,44)(21,36,23,31)(25,38,41,42)(27,40,37,39)(32,46,47,43)(33,48,35,45);(2,6)(3,8)(10,13)(14,16)(17,20,21,33,22,34,23,35)(18,25,37,47,26,41,27,32)(19,29,36,48,30,44,31,45)(24,38,39,43,28,42,40,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(25,41)(27,37)(29,44)(31,36)(32,47)(45,48);(10,13)(14,16)(17,21,22,23)(18,27,26,37)(19,31,30,36)(20,33,34,35)(24,39,28,40)(25,32,41,47)(29,45,44,48)(38,43,42,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fhi,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=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693467);