Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
549 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 127 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)(7,16)(8,13)(9,22)(10,20)(11,15)(12,18)(14,24)(17,33)(19,34)(21,31)(23,27)(25,32)(26,30)(28,36)(29,35);(5,7,11,19)(6,12,15,29)(8,17,10,26)(9,23,25,36)(13,27,20,28)(14,30,31,33)(16,32,34,22)(18,21,35,24);(2,4)(5,6)(7,12)(8,20)(9,14)(10,13)(11,15)(16,18)(17,28)(19,29)(21,32)(22,24)(23,30)(25,31)(26,27)(33,36)(34,35);(5,8,11,10)(6,13,15,20)(7,17,19,26)(9,21,25,24)(12,27,29,28)(14,22,31,32)(16,33,34,30)(18,23,35,36)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #499770, #499771, #499774, … |
| maximal normal subgroup of | #499770, #499771, #499774, … |
| maximal quotient of | #484204, #484550, #484629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26515);