Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 36 points:
(5,6,11,17)(7,18,21,33)(8,12,25,29)(9,16,10,15)(13,26,31,20)(14,24,28,22)(19,32,23,30)(27,35,34,36);(1,2)(3,4)(6,12)(8,22)(9,10)(13,27)(14,16)(15,28)(17,29)(18,32)(19,35)(20,26)(23,36)(24,25)(30,33)(31,34);(2,4)(5,7)(6,13)(8,23)(9,26)(10,20)(11,21)(12,27)(14,32)(15,33)(16,18)(17,31)(19,25)(22,36)(24,35)(28,30)(29,34)Polycyclic presentation:
2^8|a^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13321, #498830, #498832, … |
| maximal normal subgroup of | C2×#13321, #498830, #498832, … |
| maximal quotient of | #142237, #142250, #142286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6309);