Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(11,19)(12,13)(15,20)(16,17)(21,22)(23,24);(2,5)(6,8)(9,10)(11,20)(12,17)(13,16)(14,18)(15,19)(21,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15,18,23)(12,22)(13,19)(14,21)(16,24,17,20);(3,7)(6,8)Polycyclic presentation:
2^9|c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235102, #72235111, #72237047, … |
| maximal normal subgroup of | #72235102, #72235111, #72237047, … |
| maximal quotient of | C2×#382414, #72293048, #72293064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289610);