Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486115 |
| 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×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
861 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(21,31)(28,29);(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25);(2,7)(3,10)(8,12)(11,13)(17,20,22,19)(18,24,26,27)(21,29,31,28)(23,30,32,25)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72279095 |
| maximal normal subgroup of | #72279095 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393316);