Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
811 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 811 conjugacy classes of subgroups — too many to draw. Only its 143 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15)(7,16,20,17)(8,22,24,30)(9,25,11,14)(12,21,28,19)(13,26,29,23)(18,27,34,36)(31,35,32,33);(5,7,10,20)(6,12,15,28)(8,18,24,34)(9,21,11,19)(13,27,29,36)(14,17,25,16)(22,31,30,32)(23,35,26,33);(2,4)(5,6)(7,17)(8,13)(9,14)(10,15)(11,25)(12,19)(16,20)(18,32)(21,28)(22,23)(24,29)(26,30)(27,35)(31,34)(33,36);(5,8,10,24)(6,13,15,29)(7,18,20,34)(9,26,11,23)(12,27,28,36)(14,30,25,22)(16,31,17,32)(19,35,21,33)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=h,e^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,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151147, #151149, #151374, … |
| maximal normal subgroup of | #151147, #151149, #151374, … |
| maximal quotient of | #244503, #244510, #244512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16879);