Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,22)(16,20)(17,19)(18,21)(23,28)(24,27)(25,26);(5,6)(7,8)(9,10)(11,12)(13,15,19,16)(14,20,17,22)(18,24,26,23)(21,28,25,27);(1,2)(3,4)(6,10)(7,11)(13,16,19,15)(14,20,17,22)(18,24,26,23)(21,27,25,28);(5,7,9,11)(6,8,10,12)(13,17,19,14)(15,22,16,20)(18,25,26,21)(23,28,24,27);(5,8)(6,11)(7,10)(9,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482898, #6352884, #6352894, … |
| maximal normal subgroup of | #482898, #6352884, #6352894, … |
| maximal quotient of | #544422, #545009, #545360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34941);