Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,21,16,20,23,27)(15,22,25,28,24,18,26,19);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,23,26,16,24,17,25)(14,18,27,28,20,22,21,19);(6,10)(9,12)(13,15,16,24)(14,19,20,28)(17,26,23,25)(18,27,22,21);(1,3,4,2)(14,20)(18,22)(19,28)(21,27)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27772, #6457645, #6457663, … |
| maximal normal subgroup of | C2×#27772, #6457645, #6457663, … |
| maximal quotient of | #315045, #315046, #315047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20760);