Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #121 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #388 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
156 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,10)(4,7)(8,15)(9,13)(12,16)(17,18,20,23,22,26,21,25)(19,27,28,32,30,31,29,24);(1,2,7,13,5,9,4,6)(3,8,14,15,10,16,11,12)(18,23,26,25)(24,27,32,31);(1,3,4,11,5,10,7,14)(2,8,6,12,9,16,13,15)(17,19,20,28,22,30,21,29)(18,24,25,31,26,32,23,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=egh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9586, #203058, #203062, … |
| maximal normal subgroup of | C2×#9586, #203058, #203062, … |
| maximal quotient of | #101897, #101898, #101901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2574);