Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,16)(12,20)(13,22)(14,17)(15,18)(19,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,20)(14,24)(15,21)(17,19)(18,23);(2,5)(6,7)(9,12,15,19)(10,11,18,21)(13,23,14,16)(17,24,22,20);(1,3,4,8)(2,6,5,7);(9,10,15,18)(11,19,21,12)(13,17,14,22)(16,20,23,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578201);