Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
2181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,12,15,13,17,19,16)(11,14,20,18,21,23,24,22);(2,5)(3,7)(9,10)(11,18)(12,16)(13,17)(14,20)(15,19)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,20)(21,23)(22,24);(2,5)(6,8)(9,11,19,24,13,21,12,20)(10,14,16,22,17,23,15,18);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=h,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=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053015);