Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394177 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(18,21)(19,22)(23,24);(9,11)(10,13)(12,15)(14,16)(17,18,22,24,20,23,19,21);(2,4)(11,15)(13,16);(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311758, #72311762, #72311768 |
| maximal normal subgroup of | #72311758, #72311762, #72311768 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308782);