Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,11,13,10,8,9,19)(7,14,18,15,17,12,16,20);(2,4)(5,7)(6,12)(8,14)(9,16)(10,17)(11,18)(13,20)(15,19);(5,8)(6,10)(7,15)(9,19)(11,13)(12,16)(14,18)(17,20);(2,4)(5,9,10,11)(6,13,8,19)(7,16,17,18)(12,20,14,15)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24402, #6437792, #6437795, … |
| maximal normal subgroup of | C2×#24402, #6437792, #6437795, … |
| maximal quotient of | #267055, #267062, #267067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17390);