Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23);(2,6)(3,4)(7,8)(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23);(1,3,5,4)(2,6,8,7)(12,21)(14,22)(15,20)(18,24)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efg,d^c=dh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #130227, #130228, #130229, … |
| maximal normal subgroup of | #130227, #130228, #130229, … |
| maximal quotient of | #122695, #122699, #122702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4862);