Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1856 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)(3,7,8,5)(4,6)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,24,21)(22,23);(9,11)(10,14)(12,21)(13,20)(15,23)(16,18)(17,24)(19,22);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,20)(14,17)(16,24)(18,23)(19,22);(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72226017, #72226041, #72226081, … |
| maximal normal subgroup of | #72226017, #72226041, #72226081, … |
| maximal quotient of | #72226006, #72226008, #72237395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121795);