Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| 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 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
301 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(9,11,13,12)(10,14,15,16)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(2,5)(6,8)(11,12)(14,16)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=efh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30156, #6434831, #6434842, … |
| maximal normal subgroup of | C2×#30156, #6434831, #6434842, … |
| maximal quotient of | #380443, #380447, #380495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23144);