Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4249 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)(4,5)(7,8)(9,10)(11,19)(12,21)(13,16)(14,17)(15,18)(20,23)(22,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,20)(14,21)(15,22)(16,19)(17,24)(18,23);(9,12)(10,11)(13,19)(14,24)(15,23)(16,20)(17,21)(18,22);(9,13,15,14)(10,16,18,17)(11,20,22,21)(12,19,23,24)Polycyclic presentation:
2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,g^c=gi,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72312618, #72312623 |
| maximal normal subgroup of | #72312618, #72312623 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536384);