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 |
| prime radical | #3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 331 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15)(7,17,20,33)(8,23,24,13)(9,27,29,34)(11,30,28,16)(12,31,26,21)(14,32,19,18)(22,36,35,25);(5,7,18,23,11,22,29,31,10,20,32,13,28,35,9,21)(6,12,27,25,16,8,19,33,15,26,34,36,30,24,14,17);(2,4)(5,8)(6,13)(7,19)(9,25)(10,24)(11,26)(12,28)(14,20)(15,23)(16,21)(17,32)(18,33)(22,34)(27,35)(29,36)(30,31);(7,20)(8,24)(12,26)(13,23)(17,33)(21,31)(22,35)(25,36)Polycyclic presentation:
2^8|a^2=h,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 | #502143, #502144, #502147, … |
| maximal normal subgroup of | #502143, #502144, #502147, … |
| maximal quotient of | #490480, #490551, #490583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26601);