Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,13)(9,18)(10,14)(12,15)(16,19)(17,20);(1,2,3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,20)(14,19);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,12,19,20);(6,14)(11,19)(12,20)(13,18)Polycyclic presentation:
2^8|b^2=f,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #247688, #247690, #247691, … |
| maximal normal subgroup of | #247688, #247690, #247691, … |
| maximal quotient of | #6355311, #6355312, #6355313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53413);