Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C23×#23900 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
174 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136092, #136093, #136094, … |
| maximal normal subgroup of | #136092, #136093, #136094, … |
| maximal quotient of | #99315, #99316, #99317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1853);