Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,18)(11,19,21,23)(12,16,22,20)(13,17,14,24);(2,4)(9,11,14,20,15,21,13,16)(10,12,17,19,18,22,24,23);(5,7,8,6)(9,12)(10,16)(11,17)(13,23)(14,19)(15,22)(18,20)(21,24);(9,13,15,14)(10,17,18,24)(11,16,21,20)(12,19,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,g^2=h,b^a=be,e^a=eh,g^a=gh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #355850, #355852, #356105, … |
| maximal normal subgroup of | #355850, #355852, #356105, … |
| maximal quotient of | #182151, #182162, #182175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13405);