Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,17)(11,16,18,23)(12,13,22,14)(19,24,20,21);(2,5)(6,7)(9,11,14,20,15,18,13,19)(10,16,12,21,17,23,22,24);(9,12,13,17,15,22,14,10)(11,16,19,21,18,23,20,24);(1,3,4,8)(2,6,5,7)(11,18)(16,23)(19,20)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152957, #152959, C2×#28635, … |
| maximal normal subgroup of | #152957, #152959, C2×#28635, … |
| maximal quotient of | #324458, #324459, #324462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21623);