Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10);(6,9)(7,12)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(2,4)(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21);(2,4)(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632828, #6632835, #6632851, … |
| maximal normal subgroup of | #6632828, #6632835, #6632851, … |
| maximal quotient of | #340467, #340468, #340471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23868);