Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,19)(15,17)(18,22)(20,23)(21,24);(1,3,6,2)(4,8,5,7)(9,10)(11,13)(12,18)(14,19)(15,17)(16,22)(20,24)(21,23);(9,11,13,10,15,19,14,17)(12,18,21,16,20,24,22,23);(1,4,6,5)(2,7,3,8)(12,20)(16,23)(18,24)(21,22);(1,5,6,4)(2,7,3,8)(9,12,13,21,15,20,14,22)(10,16,19,24,17,23,11,18)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=f,e^2=fhi,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,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718825);