Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #537 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1181 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,29,24,26,40,43,22,27,39,45,20,34,36,47)(19,30,33,46,31,37,44,48,32,25,41,42,35,21,28,38);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,32)(18,25,27,30)(20,35,24,31)(21,34,37,26)(23,33,39,41)(28,40,44,36)(29,42,45,46)(38,47,48,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,24)(18,26,27,34)(19,31,32,35)(21,25,37,30)(23,36,39,40)(28,33,44,41)(29,43,45,47)(38,42,48,46);(2,8)(6,9)(7,16)(14,15)(18,27)(19,32)(26,34)(28,44)(29,45)(31,35)(33,41)(43,47);(17,21,23,38,24,30,40,46,22,37,39,48,20,25,36,42)(18,28,29,19,26,33,43,31,27,44,45,32,34,41,47,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cf,d^b=di,e^b=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529993);