Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
265 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9,10,11)(13,14,17,20,18,21,16,26)(15,22,25,27,23,28,24,19);(5,6,7,11,8,10,12,9)(13,15,18,23)(14,19)(16,24,17,25)(20,22)(21,27)(26,28);(2,4)(15,23)(19,27)(22,28)(24,25)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #97806, #97807, #97808, … |
| maximal normal subgroup of | #97806, #97807, #97808, … |
| maximal quotient of | #99743, #99745, #99747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2113);