Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(17,18,21,25)(19,27,29,23)(20,24,22,26)(28,32,30,31);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,15,14)(19,28)(20,22)(23,31)(24,26)(27,32)(29,30);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,29)(18,23,25,27)(20,28,22,30)(24,31,26,32);(3,7)(5,8)(11,16)(13,14)(17,20,21,22)(18,24,25,26)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#32749, #6631310, #6631321, … |
| maximal normal subgroup of | C2×#32749, #6631310, #6631321, … |
| maximal quotient of | #391234, #391238, #391309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25737);