Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 131072 |
| 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 | |
1061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,8)(7,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(5,6,9,8)(7,10,11,12)(14,18)(19,21)(20,25)(26,27);(1,2)(3,4)(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(5,7,9,11)(6,10,8,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(5,8,9,6)(7,12,11,10)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=g,b^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #112324, #201599, #235290, … |
| maximal normal subgroup of | #112324, #201599, #235290, … |
| maximal quotient of | #556354, #556361, #556379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30541);