Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #2150 | C22 |
| socle | C2 | #7063 |
| prime radical | C23 | #349 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,6,8,17,22,5,11,20,27,16,23,30,31,7,12,19,28,21,24,29,32,14,9,18,25,3,13,15,26);(1,3,14,21,7,16,5,6)(2,8,11,23,12,24,9,13)(4,17,20,30,19,29,18,15)(10,26,25,32,28,31,27,22)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=gi,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=deg,d^c=dgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 924);