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) = C26 |
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 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(6,8)(10,13)(11,15)(17,18,21,20)(19,23,24,22);(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,23)(21,24);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=eg,d^2=eh,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #100159, #100167, #100175, … |
| maximal normal subgroup of | #100159, #100167, #100175, … |
| maximal quotient of | #153808, #153824, #153879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7228);