Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #23386 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,8)(9,10,13,12,14,16,23,22)(11,15,20,18,21,24,19,17);(1,2,4,3)(9,10,13,12,14,16,23,22)(11,17,19,24,21,18,20,15);(5,6,7,8)(9,11)(10,15)(12,18)(13,20)(14,21)(16,24)(17,22)(19,23);(1,3,4,2)(9,12,13,16,14,22,23,10)(11,18,20,24,21,17,19,15)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72310050, #72310129, #72310142, … |
| maximal normal subgroup of | #72310050, #72310129, #72310142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286208);