Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
214 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19)(12,16)(13,17,22,23)(14,18)(15,20,24,21);(9,11,14,12)(10,15,18,24)(13,21,22,20)(16,23,19,17);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,16,18,19)(13,21,22,20)(15,23,24,17)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,d^b=dh,e^b=efh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94129, #94133, #94152, … |
| maximal normal subgroup of | #94129, #94133, #94152, … |
| maximal quotient of | #122377, #122385, #122393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4466);