Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
517 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,7)(6,8)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,21)(22,24);(2,5)(6,8)(10,11)(12,16)(18,20)(19,24)(25,26,28,30)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^8|a^2=ef,b^2=efgh,c^2=ef,d^2=efg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #125852, #126119, #126199, … |
| maximal normal subgroup of | #125852, #126119, #126199, … |
| maximal quotient of | #154894, #154919, #154995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9224);