Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,21,17,20,27,16)(15,22,25,26,23,19,28,24);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,19,20,22)(16,26,21,24)(18,25,27,28);(2,4)(14,20)(15,23)(16,21)(25,28);(5,8,9,7)(6,10,11,12)(13,16,27,20,17,21,18,14)(15,24,28,19,23,26,25,22)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=eg,g^2=h,b^a=be,c^a=cf,c^b=ch,d^b=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21606, #6373133, #6373136, … |
| maximal normal subgroup of | C2×#21606, #6373133, #6373136, … |
| maximal quotient of | #212540, #212541, #212542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14594);