Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
675 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 675 conjugacy classes of subgroups — too many to draw. Only its 167 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)(7,16)(8,20)(9,14)(10,25)(11,15)(12,18)(13,24)(17,33)(19,28)(21,32)(22,35)(23,31)(26,29)(27,36)(30,34);(5,7)(6,12)(8,21)(9,18)(10,26)(11,19)(13,30)(14,28)(15,29)(16,25)(17,24)(20,27)(22,36)(23,34)(31,32)(33,35);(2,4)(5,8,10,23,11,24,9,22)(6,13,14,31,15,20,25,35)(7,17,18,34,19,21,26,36)(12,27,16,32,29,30,28,33);(5,9,11,10)(6,14,15,25)(7,18,19,26)(8,22,24,23)(12,28,29,16)(13,31,20,35)(17,34,21,36)(27,33,30,32)Polycyclic presentation:
2^8|c^2=e,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,c^b=cg,e^b=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137186, #137193, #142288, … |
| maximal normal subgroup of | #137186, #137193, #142288, … |
| maximal quotient of | #238804, #238808, #238812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16508);