Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112555 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35064 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,20,15)(12,16)(13,17)(14,19,24,21)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,21)(13,20)(15,16)(17,24)(18,23)(19,22);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,21)(13,20)(14,23)(16,24)(17,18)(19,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231835 |
| maximal normal subgroup of | #72231835 |
| maximal quotient of | #72231802, #72278199 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19751);