Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1106 conjugacy classes of subgroups; only the 415 normal subgroups are shown.
This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 415 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) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(11,15)(14,16)(18,20)(19,23);(2,3)(6,7)(10,13)(14,16)(17,18,21,20)(19,24,23,22);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133176, #133178, #251731, … |
| maximal normal subgroup of | #133176, #133178, #251731, … |
| maximal quotient of | #553759, #553796, #553990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30307);