Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393975 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,16)(14,18)(15,21)(19,24)(22,23);(1,3,4,8)(2,5,6,7)(12,21)(14,23)(15,20)(18,22);(9,11)(10,14)(12,22)(13,19)(15,24)(16,23)(17,20)(18,21);(12,21)(15,20)(17,24)(18,22)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233099, #72233137, #72233147, … |
| maximal normal subgroup of | #72233099, #72233137, #72233147, … |
| maximal quotient of | #72233090, #72233844, #72264073 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301606);