Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,12,13)(11,15,16,14)(17,18,21,24)(19,23,20,22);(5,7,8,6)(10,13)(14,15)(17,19,21,20)(18,22,24,23);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,23,24,22);(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=eg,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #100200, #100202, #100216, … |
| maximal normal subgroup of | #100200, #100202, #100216, … |
| maximal quotient of | #153760, #153773, #153817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7028);