Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814805 |
| 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 | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,13,18)(11,21,22,16)(12,19,24,23)(14,17,15,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,19)(17,24)(18,21);(10,17,18,20)(11,19,22,23);(1,3,4,7)(2,6,5,8)(10,18)(12,24)(16,21)(17,20);(2,5)(6,8)(9,12,15,16,13,24,14,21)(10,19,20,11,18,23,17,22)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=fi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722318);