Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | |
1187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,11,12)(7,8,10,9)(14,15)(19,20);(5,7,11,10)(6,9,12,8)(13,14,18,15)(16,19,17,20);(1,2,3,4)(5,8,11,9)(6,7,12,10)(13,15,18,14)(16,20,17,19);(5,9,11,8)(6,7,12,10)(13,16,18,17)(14,19,15,20);(5,10,11,7)(6,9,12,8)(13,17,18,16)(14,20,15,19)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132234, #248326, #248327, … |
| maximal normal subgroup of | #132234, #248326, #248327, … |
| maximal quotient of | #517984, #517985, #518002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29050);