Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 65536 |
| 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 | |
563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,16,17)(15,19,20,18)(21,22)(23,28)(24,27)(25,26);(6,7)(8,12)(14,17)(18,19)(21,23,26,24)(22,27,25,28);(5,7,9,6)(8,11,12,10)(13,15,16,20)(14,18,17,19)(21,24,26,23)(22,27,25,28);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^8|a^2=fg,b^2=ef,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#15581, #609165, #609209, … |
| maximal normal subgroup of | C2×#15581, #609165, #609209, … |
| maximal quotient of | #154715, #154878, #154957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8569);