Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| 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 | |
569 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)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22)(23,28)(24,27)(25,26);(6,9)(10,11)(15,19)(17,20)(21,23,26,24)(22,27,25,28);(5,6)(7,11)(8,9)(10,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,23)(22,27,25,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,14)(15,17)(16,18)(19,20)(21,25,26,22)(23,28,24,27)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16412, #610924, #611032, … |
| maximal normal subgroup of | C2×#16412, #610924, #611032, … |
| maximal quotient of | #155638, #155705, #155806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9400);