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) = C24 |
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 | C22×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 240 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)(11,17,19,24)(12,20,22,15)(13,16)(14,18)(21,23);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,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 | #94128, #94132, #94155, … |
| maximal normal subgroup of | #94128, #94132, #94155, … |
| maximal quotient of | #122376, #122384, #122392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4465);