Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188035 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
4130 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)(3,7,8,5)(4,6)(9,10,13,18,14,17,12,16)(11,19,20,24,22,15,21,23);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,18)(11,20,22,21)(15,23,19,24);(9,12,14,13)(10,16,17,18)(11,21,22,20)(15,24,19,23)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237350, #72237351, #72237426, … |
| maximal normal subgroup of | #72237350, #72237351, #72237426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95285);