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 | |
575 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 575 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,14)(9,22)(10,26)(11,16)(12,18)(15,24)(17,30)(19,32)(20,31)(21,33)(23,27)(25,36)(28,35)(29,34);(5,7)(6,12)(8,17)(9,23)(10,20)(11,19)(13,22)(14,27)(15,31)(16,29)(18,24)(21,34)(25,35)(26,28)(30,33)(32,36);(2,4)(5,7)(6,13)(8,20)(9,18)(10,17)(11,19)(12,22)(14,31)(15,27)(16,32)(21,35)(23,24)(25,34)(26,30)(28,33)(29,36);(5,8,11,10)(6,14,16,26)(7,17,19,20)(9,21,25,24)(12,27,29,28)(13,30,32,31)(15,22,33,36)(18,23,34,35)Polycyclic presentation:
2^8|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 | #499769, #499772, #499773, … |
| maximal normal subgroup of | #499769, #499772, #499773, … |
| maximal quotient of | #484203, #484549, #484628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26514);