Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | Aut(G) = C23×#2403 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| 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 | C23×C4 | C23 |
| prime radical | #3194 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
232 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,7,10)(3,6,9,5)(4,11)(8,12)(13,15)(14,16)(17,18)(19,22,26,30)(20,27,28,23)(21,24)(25,29)(31,32);(1,3,5,2,7,9,6,10)(4,11,13,8,14,16,12,15)(17,19,21,26)(18,22)(20,25,28,31)(23,29)(24,30)(27,32);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9332, #199518, #199520, … |
| maximal normal subgroup of | C2×#9332, #199518, #199520, … |
| maximal quotient of | #99997, #99998, #100001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2320);