Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486811 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,16)(4,12)(5,24)(6,11)(7,14)(8,26)(9,15)(10,18)(13,21)(17,27)(19,32)(20,30)(22,28)(23,29)(25,31);(1,3,7,19)(2,10,14,12)(4,21,18,5)(6,8,20,9)(11,27,30,31)(13,32,24,16)(15,28,26,29)(17,22,25,23);(2,11)(4,22)(5,17)(10,27)(12,31)(13,28)(14,30)(15,16)(18,23)(21,25)(24,29)(26,32);(1,4,8,22)(2,12,26,28)(3,17,20,5)(6,21,19,25)(7,18,9,23)(10,15,29,14)(11,13,32,31)(16,27,30,24)Polycyclic presentation:
2^9|b^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72234735, #72234736, #72234738 |
| maximal normal subgroup of | #72234735, #72234736, #72234738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394632);