Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3047 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)(9,10,14,19,15,20,13,18)(11,21,22,12,17,24,23,16);(1,2)(3,6)(4,5)(7,8)(9,11,15,17)(10,16,20,12)(13,22,14,23)(18,21,19,24);(2,5)(6,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,22)(23,24);(2,5)(6,8)(9,12,15,16)(10,17,20,11)(13,21,14,24)(18,22,19,23);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,22,17,23)(12,24,16,21)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398633);