Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
1047 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)(3,8)(4,7)(5,10)(6,11)(9,16)(12,21)(13,19)(14,23)(15,24)(17,20)(18,26)(22,28)(25,30)(27,31)(29,32);(2,7)(4,16)(8,19)(9,10)(11,20)(13,28)(17,30)(21,31)(22,23)(24,32)(25,26)(27,29);(3,12)(4,17)(8,21)(13,28)(14,15)(16,30)(19,31)(22,32)(23,24)(27,29);(1,3,5,14)(2,8,10,23)(4,13,16,28)(6,15,18,12)(7,19,9,22)(11,24,26,21)(17,29,30,27)(20,32,25,31)Polycyclic presentation:
2^9|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72277997, #72284184, #72284207, … |
| maximal normal subgroup of | #72277997, #72284184, #72284207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394501);