Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(23,27)(25,28);(2,4)(6,9)(10,12)(14,18)(15,20)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,19,18,16)(21,22)(23,25)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,20,18,15)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=ef,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 | C2×#14941, #607988, #608076, … |
| maximal normal subgroup of | C2×#14941, #607988, #608076, … |
| maximal quotient of | #154138, #154220, #154227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7929);