Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14696 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,8)(3,13,17,21,16,9,15,24)(4,10,20,25,19,26,18,22)(14,23,30,32,29,27,28,31);(2,8,12,11)(9,21,13,24)(10,22,26,25)(23,31,27,32);(1,3,5,15,7,16,6,17)(2,9,11,24,12,13,8,21)(4,14,18,28,19,29,20,30)(10,23,25,32,26,27,22,31);(1,4)(2,10)(3,14)(5,18)(6,20)(7,19)(8,22)(9,23)(11,25)(12,26)(13,27)(15,28)(16,29)(17,30)(21,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^c=fi,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72219684, #72219692, … |
| maximal normal subgroup of | #72215931, #72219684, #72219692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300022);