Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1188 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)(3,7)(4,5)(6,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,20,22)(21,24);(2,5)(6,7)(9,11)(10,14)(12,20,21,18)(13,19)(15,23,24,22)(16,17);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20);(3,8)(6,7)(14,17)(15,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72318333, #72318408, #72318437, … |
| maximal normal subgroup of | #72318333, #72318408, #72318437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387051);