Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #1514 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1514 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | #469 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 45 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)(3,13,15,30)(4,18,20,25)(5,11)(6,9)(7,12)(8,14)(10,21,27,22)(16,26,17,29)(19,28)(23,32)(24,31);(1,3)(2,8)(4,19)(5,17)(6,16)(7,15)(9,23)(10,26)(11,28)(12,24)(13,18)(14,22)(20,32)(21,31)(25,30)(27,29);(2,9,12,11)(4,20)(8,23,24,28)(10,18,27,25)(13,29,30,26)(14,31)(19,32)(21,22)Polycyclic presentation:
2^8|a^2=e,c^2=fg,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,d^c=dh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11836, #481946, #481947, … |
| maximal normal subgroup of | C2×#11836, #481946, #481947, … |
| maximal quotient of | #124541, #124542, #124633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4824);