Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
293 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11)(21,22)(23,26,30,34)(24,27)(25,28)(29,33,31,35)(32,36);(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20)(21,23)(22,26,28,34)(24,31)(25,30)(27,35,36,33)(29,32);(2,4)(7,15)(11,19)(14,20)(16,17)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,33,34,35)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,f^2=gh,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 | #97301, #97302, #97303, … |
| maximal normal subgroup of | #97301, #97302, #97303, … |
| maximal quotient of | #99741, #99745, #99756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2141);