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 | C23×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23);(2,4)(9,11,13,21,14,22,12,20)(10,15,16,23,18,24,17,19);(5,7,8,6)(9,10)(11,15)(12,17)(13,16)(14,18)(19,20)(21,23)(22,24);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,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 | #355851, #355853, #356104, … |
| maximal normal subgroup of | #355851, #355853, #356104, … |
| maximal quotient of | #182150, #182163, #182174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13404);