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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
565 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,16,20,19)(21,22)(23,24)(25,27)(26,28);(7,11)(10,12)(14,18)(15,20)(21,23,25,28)(22,24,27,26);(5,7)(6,10)(8,12)(9,11)(13,15,17,20)(14,19,18,16)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,20,18,15)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^8|a^2=efg,b^2=ef,c^2=e,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14909⋊C3 |
| maximal subgroup of | C2×#14909, #607918, #607922, … |
| maximal normal subgroup of | C2×#14909, #607918, #607922, … |
| maximal quotient of | #154127, #154293, #154297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7897);