Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3537 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
531 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:
(1,2,6,11)(3,13,16,29)(4,10,5,20)(7,12,23,28)(8,21,26,14)(9,22,18,19)(15,24,17,31)(25,30,27,32);(1,3)(2,8)(4,14)(5,21)(6,16)(7,17)(9,24)(10,29)(11,26)(12,27)(13,20)(15,23)(18,31)(19,30)(22,32)(25,28);(1,2)(3,8)(4,18)(5,9)(6,11)(7,12)(10,22)(13,30)(14,31)(15,25)(16,26)(17,27)(19,20)(21,24)(23,28)(29,32);(1,4,6,5)(2,9,11,18)(3,14,16,21)(7,19,23,22)(8,24,26,31)(10,28,20,12)(13,27,29,25)(15,32,17,30)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244500, #244501, #244506, … |
| maximal normal subgroup of | #244500, #244501, #244506, … |
| maximal quotient of | #169618, #169636, #169681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11407);