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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18)(19,26,27,30)(20,23,28,29)(21,24)(22,25)(31,32);(1,3,6,12)(2,7)(4,11,5,13)(8,15)(9,10)(14,16)(17,19,21,29,22,27,31,23)(18,20,24,30,25,28,32,26);(2,8)(4,5)(7,15)(11,13)(17,20)(18,23)(19,24)(21,30)(22,28)(25,29)(26,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=g,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 | #132609, #132610, #132611, … |
| maximal normal subgroup of | #132609, #132610, #132611, … |
| maximal quotient of | #100127, #100129, #100131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2437);