Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2410 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,5,6,23)(3,12,15,9,17,26,16,14)(4,18,21,20,22,28,24,10)(13,30,31,27,32,25,19,29);(1,3)(2,9)(4,19)(5,14)(6,16)(7,15)(8,17)(10,27)(11,26)(12,23)(13,21)(18,25)(20,29)(22,31)(24,32)(28,30);(1,4)(2,10)(3,13)(5,20)(6,21)(7,24)(8,22)(9,25)(11,28)(12,29)(14,30)(15,19)(16,31)(17,32)(18,23)(26,27);(1,5,8,2)(3,14,17,9)(4,20,22,10)(6,11,7,23)(12,16,26,15)(13,30,32,25)(18,21,28,24)(19,29,31,27)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24473, #6437949, #6437951, … |
| maximal normal subgroup of | C2×#24473, #6437949, #6437951, … |
| maximal quotient of | #268288, #268290, #268291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17461);