Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C42 |
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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,5,6,7)(9,10)(11,15)(12,19,20,14)(13,16)(17,23)(18,22,24,21);(1,2)(3,7)(4,6)(5,8)(10,14,16,19)(11,17)(12,20)(15,22,23,21);(9,11,13,17)(10,15,16,23)(12,18,20,24)(14,21,19,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efgh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #129825, #129828, #129830, … |
| maximal normal subgroup of | #129825, #129828, #129830, … |
| maximal quotient of | #124091, #124123, #124196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4705);