Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,17)(16,18)(19,20);(1,2,3,4)(14,17)(18,19);(14,18)(16,20)(17,19);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237132, #72237137, #72237141, … |
| maximal normal subgroup of | #72237132, #72237137, #72237141, … |
| maximal quotient of | #72250018, #72250272, #72256842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299356);