Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533135 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,18,15,12)(13,16)(14,17)(19,24,22,20)(21,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,22)(23,24);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,18)(16,22)(19,23)(21,24);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,17,23)(11,19,15,22)(12,20,18,24);(3,7)(6,8)(13,21)(16,23)(19,22)(20,24)Polycyclic presentation:
2^9|a^2=fg,d^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268363, #72268427, #72268451, … |
| maximal normal subgroup of | #72268363, #72268427, #72268451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449434);