Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499780 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(18,21)(19,23)(22,24);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(18,22)(21,24);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,24,23,21);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(18,22)(21,24);(10,14)(13,15)Polycyclic presentation:
2^9|c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282041, #72305294, #72305306 |
| maximal normal subgroup of | #72282041, #72305294, #72305306 |
| maximal quotient of | #72305627, #72305738, #72315978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6408743);