Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501052 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,15,27,26,37,25,17,14,20,12,18,13,19,29,36)(11,16,28,30,23,32,38,35,24,34,22,31,21,33,39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,22)(13,21)(14,24)(15,28)(17,35)(18,31)(19,33)(20,34)(23,26)(25,38)(27,30)(29,39)(32,37)(36,40);(1,4,5,3)(2,6,7,8)(10,17)(11,21)(12,25)(13,26)(15,29)(16,30)(18,19)(20,36)(22,28)(23,24)(27,37)(31,34)(32,40)(33,35);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=fi,b^2=e,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408645);