Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,18)(12,17)(13,22)(14,16)(15,21)(19,24)(20,23);(1,3,6,2)(4,8,5,7)(9,11)(10,15,16,23)(12,20)(13,21)(14,19)(17,18,22,24);(2,3)(7,8)(10,16)(11,19)(17,22)(20,21);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,17,16,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eh,g^a=ghi,c^b=ch,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72235203 |
| maximal normal subgroup of | #72235203 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88866);