Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,15)(12,19,20,14)(13,16)(17,23,24,21)(18,22);(10,14)(15,21)(16,19)(22,23);(2,5)(6,8);(9,11,13,18)(10,15,16,22)(12,17,20,24)(14,21,19,23)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,f^b=fg,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237533, #72237555, #72237556, … |
| maximal normal subgroup of | #72237533, #72237555, #72237556, … |
| maximal quotient of | #72284451, C2×#480245, #72292794 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387441);