Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,27,24)(23,28,34,35)(25,29,26,30)(31,33,36,32);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,27,34)(22,28,24,35)(25,32,26,33)(29,31,30,36);(2,4)(21,24,27,22)(23,31,34,36)(25,29,26,30)(28,33,35,32);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,26)(22,29,24,30)(23,32,34,33)(28,31,35,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,33)(25,26)(28,36)(29,30)(31,35)(32,34)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584782);