Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,17,15)(16,19,20,18)(22,24)(23,27);(6,9)(10,12)(14,15)(18,19)(21,22,25,24)(23,26,27,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,28)(25,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,20)(14,18,15,19)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^8|a^2=ef,b^2=gh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #101153, #101154, #101169, … |
| maximal normal subgroup of | #101153, #101154, #101169, … |
| maximal quotient of | #154161, #154273, #154360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8086);