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 |
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 | |
315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,13,15)(11,14,12,16)(17,18,20,21)(19,23,24,22);(2,5)(3,8)(9,11,13,12)(10,14,15,16)(18,21)(19,22)(23,24);(9,12,13,11)(10,14,15,16)(17,19,20,24)(18,22,21,23);(1,3,4,8)(2,6,5,7)(10,15)(14,16)(17,18,20,21)(19,22,24,23)Polycyclic presentation:
2^8|a^2=efh,b^2=fh,c^2=f,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 | #6642927, #6642937, #6642961, … |
| maximal normal subgroup of | #6642927, #6642937, #6642961, … |
| maximal quotient of | #384633, #384731, #384790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25472);