Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487627 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,21)(22,24);(2,5)(3,8)(6,7)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(3,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(2,6)(5,7)(10,16)(11,18)(14,17)(19,24)Polycyclic presentation:
2^9|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283435 |
| maximal normal subgroup of | #72283435 |
| maximal quotient of | #72284330, #72284486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394823);