Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2323 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)(3,5)(4,6)(7,8)(9,10,15,20,16,12,14,19)(11,13,18,21,23,24,22,17);(1,2)(3,7)(4,6)(5,8)(9,11,16,23)(10,17,12,21)(13,20,24,19)(14,18,15,22);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,21)(13,22)(14,20)(15,19)(17,23)(18,24);(3,8)(5,7)(9,13,15,21,16,24,14,17)(10,18,20,23,12,22,19,11);(2,6)(5,7)(9,14,16,15)(10,19,12,20)(11,18,23,22)(13,21,24,17)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6904004);