Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(3,8)(5,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,6)(5,7);(11,18)(14,22)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224136, #72225697, #72225713, … |
| maximal normal subgroup of | #72224136, #72225697, #72225713, … |
| maximal quotient of | #72251088, #72251338, #72284533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281324);