Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | C22⋊#401 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
825 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 825 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,20,19,16)(21,22)(23,26)(24,28)(25,27);(7,11)(9,12)(14,18)(15,19)(21,23,25,28)(22,26,27,24);(5,7)(6,9)(8,11)(10,12)(13,15,17,19)(14,16,18,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,18)(17,20)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=efg,b^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | (C22⋊#401)⋊C3 |
| maximal subgroup of | C2×C22⋊#401, #607794, #607821, … |
| maximal normal subgroup of | C2×C22⋊#401, #607794, #607821, … |
| maximal quotient of | #154101, #154201, #154216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7815);