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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,18,21)(15,22,16,19)(17,20,25,28)(23,27,24,26);(5,6)(7,8)(9,10)(11,12)(13,15,18,16)(14,19,21,22)(17,24,25,23)(20,27,28,26);(1,2,3,4)(6,10)(7,11)(13,16,18,15)(14,19,21,22)(17,24,25,23)(20,26,28,27);(5,7,9,11)(6,8,10,12)(13,14)(15,19)(16,22)(17,20)(18,21)(23,26)(24,27)(25,28);(5,8)(6,11)(7,10)(9,12)(13,17,18,25)(14,20,21,28)(15,23,16,24)(19,26,22,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,d^2=h,e^2=f,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 | #482791, #6353208, #6353218, … |
| maximal normal subgroup of | #482791, #6353208, #6353218, … |
| maximal quotient of | #544548, #545099, #545540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34980);