Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,16)(11,17,19,23)(12,15,21,24)(14,20,22,18);(9,11)(10,14)(12,20,21,18)(13,19)(15,23,24,17)(16,22);(2,4)(5,7,8,6)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,b^a=bd,c^a=ce,g^a=gh,d^b=dh,h^b=hi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72317654, #72317678, #72317684, … |
| maximal normal subgroup of | #72317654, #72317678, #72317684, … |
| maximal quotient of | #72317818, #72319505 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54510);