Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112870 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,22,20)(21,24);(2,5)(3,7)(9,11)(10,14)(12,20)(13,19)(15,23)(16,22)(17,24)(18,21);(2,5)(3,7)(14,22)(17,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239024, #72279653, #72279844 |
| maximal normal subgroup of | #72239024, #72279653, #72279844 |
| maximal quotient of | #72257857, #72257858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20067);