Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| 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 | C24 | C25 |
| socle | C22 | C22×#120 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4466 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)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(2,5)(6,8);(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,e^2=f,b^a=bf,c^a=cg,d^a=dh,h^c=hi,g^d=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257824, #72260626, #72278320, … |
| maximal normal subgroup of | #72257824, #72260626, #72278320, … |
| maximal quotient of | #72261082, #72264517, #72264649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536583);