Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator, socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,23,19,21,26,28,17,22,24,27,16,15,20,25);(1,2,3,4)(6,9)(10,11)(13,15,24,28,19,14,20,27,17,21,18,25,16,22,26,23);(5,7,8,12)(6,10,9,11)(13,16,17,19)(14,21,22,15)(18,20,24,26)(23,28,27,25)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #93006, #93010, #93024, … |
| maximal normal subgroup of | #93006, #93010, #93024, … |
| maximal quotient of | #141843, #141850, #141862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6252);