Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator, socle | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,20,24,29)(19,23,28,31,26,27,30,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,30)(25,31)(29,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,26)(23,27)(28,30)(31,32);(17,20,21,29,22,18,24,25)(19,27,28,32,26,23,30,31)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=e,d^2=g,g^2=h,b^a=be,c^a=cf,c^b=ch88 conjugacy classes — view the full character table.
| Fitting subgroup of | #21668⋊C3 |
| maximal subgroup of | #496071, #496072, #496075, … |
| maximal normal subgroup of | #496071, #496072, #496075, … |
| maximal quotient of | #354387, #354521, #355041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14656);