Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
753 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 753 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(6,8)(9,10)(11,13)(12,15)(14,16);(1,2,3,4);(5,6,7,8);(10,13)(11,16)(14,15)Polycyclic presentation:
2^8|b^2=e,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #33160⋊C3, #72228315, #72228318 |
| maximal subgroup of | #100311, #100314, #100343, … |
| maximal normal subgroup of | #100311, #100314, #100343, … |
| maximal quotient of | #351638, #351640, #351642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26148);