Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487659 |
| 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 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1081 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)(5,9)(6,11)(10,16)(12,23)(13,28)(14,20)(15,25)(17,22)(18,21)(19,26)(24,30)(27,31)(29,32);(2,7)(3,12)(4,16)(8,22)(9,10)(11,21)(13,15)(14,29)(17,30)(18,31)(20,23)(24,28)(25,32)(26,27);(1,3)(2,8)(4,17)(5,15)(6,12)(7,20)(9,25)(10,24)(11,23)(13,19)(14,18)(16,29)(21,22)(26,28)(27,32)(30,31);(2,9)(3,13)(7,10)(8,23)(11,26)(12,15)(14,29)(17,30)(20,22)(21,27)(24,32)(25,28)Polycyclic presentation:
2^9|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394855);