Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(2,3)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,21)(20,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(1,3,5,2)(4,6,8,7)(9,12,13,20)(10,15,16,23)(11,18,19,24)(14,21,17,22);(1,4,5,8)(2,6,3,7)(12,20)(15,23)(18,24)(21,22)Polycyclic presentation:
2^8|c^2=e,d^2=eh,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=de37 conjugacy classes — view the full character table.
| Fitting subgroup of | #29711⋊C3 |
| maximal subgroup of | #487461, #487464, #487465, … |
| maximal normal subgroup of | #487461, #487464, #487465, … |
| maximal quotient of | #328823, #328824, #328825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22699);