Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235097 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #34738 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,19)(11,20,21,24)(12,18,13,17)(15,23,16,22);(5,8)(9,11)(10,15)(12,23)(13,22)(14,21)(16,19)(17,24)(18,20);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,23)(13,22)(14,21)(15,19)(17,20)(18,24);(11,21)(15,16)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280967, #72303526, #72303528, … |
| maximal normal subgroup of | #72280967, #72303526, #72303528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142294);