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) = C22×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 | |
539 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,23,24,22)(26,29)(27,31);(2,5)(6,8)(18,20)(19,24)(25,26,28,29)(27,30,31,32);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,23,24,22)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=efg,b^2=efh,c^2=g,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 | #125860, #126025, #126137, … |
| maximal normal subgroup of | #125860, #126025, #126137, … |
| maximal quotient of | #154897, #154963, #155031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9228);