Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2416 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,21,9)(3,12,15,7,17,13,16,23)(4,18,19,8,20,24,32,25)(14,28,29,26,31,22,30,27);(1,3,5,15,6,17,21,16)(2,7,9,12,11,23,10,13)(4,14,19,29,20,31,32,30)(8,22,18,26,25,28,24,27);(1,2)(3,13)(4,8)(5,9)(6,11)(7,15)(10,21)(12,17)(14,27)(16,23)(18,19)(20,25)(22,29)(24,32)(26,31)(28,30);(1,4,5,19,6,20,21,32)(2,8,9,18,11,25,10,24)(3,14,15,29,17,31,16,30)(7,22,12,26,23,28,13,27)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24808, #6443060, #6443061, … |
| maximal normal subgroup of | C2×#24808, #6443060, #6443061, … |
| maximal quotient of | #277851, #277854, #277857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17796);