Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×#117 | C22 |
| socle | C22 | #342 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
161 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,21,31)(12,23,24,16)(13,26)(14,18)(15,20)(17,27)(22,36,35,32)(25,30,38,37)(28,40)(29,34)(33,39);(9,11,13,22,14,21,27,35)(10,15,17,28,18,29,26,39)(12,20,25,33,24,34,38,40)(16,19,30,32,23,31,37,36)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,e^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93900, #93907, #93914, … |
| maximal normal subgroup of | #93900, #93907, #93914, … |
| maximal quotient of | #92968, #92970, #92976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 178);