Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| 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 | |
3301 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)(11,19,21,15)(12,20,23,16)(13,18)(14,17)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,22,23,18)(13,20,24,16);(2,5)(6,8)(9,12)(10,16)(11,18)(13,19)(14,23)(15,24)(17,20)(21,22);(1,3)(2,6)(4,7)(5,8)(10,17)(13,24)(15,19)(16,20);(3,7)(6,8)(9,13,14,24)(10,18,17,22)(11,20,21,16)(12,19,23,15)Polycyclic presentation:
2^9|a^2=fg,b^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317652, #72319407 |
| maximal normal subgroup of | #72317652, #72319407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449610);