Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,21)(13,17)(14,18)(15,19)(20,23)(22,24);(1,2)(3,6)(4,8)(5,7)(9,11,15,22)(10,16,19,24)(12,17,23,18)(13,20,14,21);(9,12,15,23)(10,11,19,22)(13,16,14,24)(17,20,18,21);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,19,18)(11,20,22,21)(12,16,23,24);(1,4,5,3)(2,6,7,8)(9,10)(11,12)(13,18)(14,17)(15,19)(16,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,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,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583726);