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 | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(21,31)(28,29);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(17,20,22,19)(18,24,26,27)(21,29,31,28)(23,30,32,25)Polycyclic presentation:
2^9|b^2=i,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 | #72284163, #72284184, #72284207, … |
| maximal normal subgroup of | #72284163, #72284184, #72284207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394504);