Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2293 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)(9,10,12,17,13,16,21,15)(11,14,18,23,20,22,19,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,20)(15,24)(16,22)(17,23)(19,21);(10,15)(11,18)(12,21)(16,17)(19,20)(23,24);(1,3,4,7)(2,6,5,8);(10,16)(14,22)(15,17)(23,24)Polycyclic presentation:
2^9|a^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,h^b=hi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320791 |
| maximal normal subgroup of | #72320791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339556);