Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,17)(16,21)(18,23)(19,22)(20,24);(2,5)(6,8)(9,11)(10,14)(12,20)(13,18)(15,24)(16,22)(17,23)(19,21);(10,15)(12,21)(14,22)(16,17)(19,20)(23,24);(1,3,4,7)(2,6,5,8);(11,18)(14,23)(19,20)(22,24)Polycyclic presentation:
2^9|d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,e^b=ei,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252473, #72252486, #72252490, … |
| maximal normal subgroup of | #72252473, #72252486, #72252490, … |
| maximal quotient of | #72282161, C2×#6432083 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339279);