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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(22,23)(24,28);(6,8)(7,11)(15,19)(17,20)(21,22,25,23)(24,27,28,26);(5,7,9,11)(6,10,8,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,14)(15,17)(16,18)(19,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^8|a^2=efg,b^2=f,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×#15580, #609165, #609208, … |
| maximal normal subgroup of | C2×#15580, #609165, #609208, … |
| maximal quotient of | #154714, #154874, #154894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8568);