Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,12)(9,16)(10,15)(11,17)(13,21)(14,23)(19,32)(20,30)(22,31)(24,27)(25,28)(26,36)(29,35)(33,34);(6,12)(7,19)(8,23)(9,10)(13,29)(14,15)(16,27)(17,28)(18,33)(20,35)(21,22)(24,25)(26,31)(30,32);(2,4)(5,7)(6,13)(8,20)(9,19)(10,21)(11,22)(12,26)(14,30)(15,31)(16,18)(17,32)(23,34)(24,36)(25,35)(27,29)(28,33)Polycyclic presentation:
2^8|d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93772, #93774, #151176, … |
| maximal normal subgroup of | #93772, #93774, #151176, … |
| maximal quotient of | #142252, #142280, #142364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6301);