Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2669 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2669 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,13,25,12)(3,14,15,26,17,32,16,8)(4,18,20,29,22,28,21,9)(5,10,23,30,24,31,19,27);(1,3,7,17)(2,8,13,26)(4,19,22,23)(5,20,24,21)(6,16,25,15)(9,27,29,30)(10,18,31,28)(11,32,12,14);(1,4)(2,9)(3,5)(6,20)(7,22)(8,10)(11,18)(12,28)(13,29)(14,30)(15,23)(16,19)(17,24)(21,25)(26,31)(27,32);(1,5)(2,10)(3,4)(6,19)(7,24)(8,9)(11,27)(12,30)(13,31)(14,28)(15,21)(16,20)(17,22)(18,32)(23,25)(26,29)Polycyclic presentation:
2^8|a^2=e,b^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632110, #6632111, #6632114, … |
| maximal normal subgroup of | #6632110, #6632111, #6632114, … |
| maximal quotient of | #332841, #332842, #332847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23486);