Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1275 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(10,13)(14,15)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,15,16,14);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(20,29)(23,28)(25,32)(27,30);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=e,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355518);