Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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, 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 | |
307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,22,27,31)(20,23,25,29)(26,30,28,32);(2,5)(6,7)(17,19,25,26,21,27,20,28)(18,22,29,30,24,31,23,32);(9,11,12,16)(10,13,14,15)(19,26)(20,25)(22,30)(23,29)(27,28)(31,32);(1,3,4,8)(2,6,5,7)(19,27)(22,31)(26,28)(30,32)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28807, #6463256, #6463261, … |
| maximal normal subgroup of | C2×#28807, #6463256, #6463261, … |
| maximal quotient of | #362199, #363817, #367575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21795);