Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | Aut(G) order 8192 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 119 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)(6,11)(8,12)(9,19)(13,21)(14,22)(15,29)(17,25)(18,31)(20,26)(23,24)(27,36)(28,34)(30,35)(32,33);(5,6,12,24,9,15,20,22,10,14,26,29,19,23,8,11)(7,13,25,33,18,27,30,34,16,28,35,36,31,32,17,21);(2,4)(5,7)(6,13)(8,17)(9,18)(10,16)(11,21)(12,25)(14,28)(15,27)(19,31)(20,30)(22,34)(23,32)(24,33)(26,35)(29,36);(6,14)(7,16)(11,22)(15,23)(17,30)(18,31)(24,29)(25,35)Polycyclic presentation:
2^8|b^2=eg,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #502141, #502142, #502145, … |
| maximal normal subgroup of | #502141, #502142, #502145, … |
| maximal quotient of | #490479, #490550, #490582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26600);