Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
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 | |
739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,12,13)(11,15,16,14)(17,18,21,23)(19,20,24,22);(2,4)(10,13)(14,15)(17,19,21,24)(18,22,23,20);(1,2)(3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(9,11,12,16)(10,14,13,15)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=eh,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14148, #514589, #514591, … |
| maximal normal subgroup of | C2×#14148, #514589, #514591, … |
| maximal quotient of | #153779, #153783, #153785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7136);