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) = C22×C4 |
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 | |
627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(6,9)(7,11)(13,14)(15,18)(16,17)(19,20)(21,22,25,26)(23,28,27,24);(1,2,4,3)(6,9)(10,12)(14,17)(18,20)(22,26)(23,27);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,27)(22,24,26,28);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^8|a^2=efg,b^2=fh,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #295284, #295294, #295299, … |
| maximal normal subgroup of | #295284, #295294, #295299, … |
| maximal quotient of | #154154, #154209, #154276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8056);