Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | Aut(G) = #72306071 (order 2048) · Inn(G) = #1519 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1519 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 188 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13)(4,17,21,7)(5,10)(6,12)(8,14)(9,18,23,31)(11,20)(15,25,19,24)(16,26)(22,32,28,29)(27,30);(2,7)(3,14)(4,18)(5,20)(8,24)(9,11)(10,23)(12,17)(13,28)(15,19)(16,30)(21,31)(22,26)(25,27);(1,3,5,14,6,16,20,30)(2,8,11,13,12,27,10,26)(4,19,18,32,21,15,31,29)(7,22,23,25,17,28,9,24)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,d^c=dh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130232, #130233, #130288, … |
| maximal normal subgroup of | #130232, #130233, #130288, … |
| maximal quotient of | #122694, #122695, #122696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4855);