Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17,20,34)(8,15,10,13)(9,22,25,30)(12,27,29,19)(14,21,32,24)(18,35,26,31)(23,33,36,28);(1,2,3,4);(5,7)(6,12)(8,18)(9,23)(10,26)(11,20)(13,28)(14,31)(15,33)(16,29)(17,22)(19,24)(21,27)(25,36)(30,34)(32,35);(5,8,11,10)(6,13,16,15)(7,18,20,26)(9,21,25,24)(12,28,29,33)(14,30,32,22)(17,31,34,35)(19,23,27,36)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136904, #136908, #136937, … |
| maximal normal subgroup of | #136904, #136908, #136937, … |
| maximal quotient of | #490431, #490456, #490497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26571);