Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = C22×#72219693 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C23 | #117 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
188 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(17,18,20,23)(19,25,26,22)(21,29,28,24)(27,30,32,31);(2,7,10,9)(8,14,16,15)(17,19,21,27,20,26,28,32)(18,22,24,30,23,25,29,31);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(19,26)(22,25)(27,32)(30,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eh,d^2=g,e^2=g,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #140163, C2×#8246, #182881, … |
| maximal normal subgroup of | #140163, C2×#8246, #182881, … |
| maximal quotient of | #96765, #96767, #96770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1234);