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 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,17,21)(15,22,18,19)(16,24,26,27)(20,23,28,25);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,18)(14,19,21,22)(16,25,26,23)(20,24,28,27);(2,4)(5,8,9,12)(6,10,7,11)(14,19)(15,18)(20,27)(21,22)(23,25)(24,28);(6,7)(10,11)(13,16,17,26)(14,20,21,28)(15,23,18,25)(19,27,22,24)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6633228, #6633235, #6633251, … |
| maximal normal subgroup of | #6633228, #6633235, #6633251, … |
| maximal quotient of | #337724, #337726, #337732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23732);