Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7402 |
| automorphism | Aut(G) = C22×#501097 (order 2048) · Inn(G) = #353 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C2×C42 | C2×C4 |
| Frattini, cosocle | #391 | C22 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,16,13,17,19,24)(12,15,18,23,21,14,20,22);(2,5)(3,8)(6,7)(10,14)(11,18)(12,21)(15,17)(19,20)(22,23)Polycyclic presentation:
2^8|a^2=d,c^2=f,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=defh,e^b=eh,d^c=dgh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7404, #150188, #150189, … |
| maximal normal subgroup of | C2×#7404, #150188, #150189, … |
| maximal quotient of | #93215, #93219, #93231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 392);