Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
325 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,6,7)(3,5,8,4)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,20,22)(21,24);(1,3,5,7,6,8,4,2)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24);(1,4,6,5)(2,3,7,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=gh,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105652, #105653, #106194, … |
| maximal normal subgroup of | #105652, #105653, #106194, … |
| maximal quotient of | #95075, #95077, #95087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 972);