Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35528 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1592 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)(11,16,20,14)(12,15)(13,17)(18,22)(19,24,21,23);(2,5)(3,7)(9,11)(10,14)(12,21)(13,20)(15,24)(16,17)(18,23)(19,22);(10,15)(12,22)(14,23)(16,24)(17,18)(19,21);(1,3,4,7)(2,6,5,8)(9,11)(10,16,17,14)(12,19)(13,20)(15,24,18,23)(21,22)Polycyclic presentation:
2^9|a^2=gh,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318131, #72318259, #72318294 |
| maximal normal subgroup of | #72318131, #72318259, #72318294 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358237);