Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10,13,17,14,12,21,16)(11,15,20,23,19,18,24,22);(1,2,3,4)(5,6,8,7)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(5,7,8,6)(9,12)(10,14)(11,18)(13,16)(15,19)(17,21)(20,22)(23,24);(11,19)(15,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239936 |
| maximal normal subgroup of | #72239936 |
| maximal quotient of | #72239943 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260705);