Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
258 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,21,26)(18,22)(20,25,27,31)(23,28)(24,29)(30,32);(1,4,6,14)(2,7,8,12)(3,10,11,16)(5,15,9,13)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132845, #132847, #132849, … |
| maximal normal subgroup of | #132845, #132847, #132849, … |
| maximal quotient of | #100128, #100129, #100136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2434);