Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | Aut(G) = C2×#72286737 (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 | #3535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 469 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,22)(20,27)(23,31)(24,28)(26,29)(30,32);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,21,24)(19,23,25,32)(20,28,27,22)(26,30,29,31);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,27)(28,31);(19,25)(23,32)(26,29)(30,31)Polycyclic presentation:
2^8|a^2=gh,b^2=g,e^2=g,b^a=be,c^a=cf,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 | #346503, #346505, #346506, … |
| maximal normal subgroup of | #346503, #346505, #346506, … |
| maximal quotient of | #166299, #166315, #166356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11223);