Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2073 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,29,5,11,7,13)(3,14,27,33,16,9,18,42)(4,10,21,37,22,35,28,43)(6,12,30,58,23,38,26,41)(15,44,47,62,48,31,51,60)(17,45,55,39,49,32,24,57)(19,52,50,63,53,34,54,61)(20,36,25,40,46,59,56,64);(1,3,5,16)(2,9,11,14)(4,19,22,53)(6,24,23,55)(7,27,8,18)(10,34,35,52)(12,39,38,57)(13,42,29,33)(15,46,48,20)(17,30,49,26)(21,54,28,50)(25,51,56,47)(31,59,44,36)(32,58,45,41)(37,63,43,61)(40,62,64,60);(1,4)(2,10,11,35)(3,15)(5,22)(6,25)(7,28)(8,21)(9,31,14,44)(12,40,38,64)(13,43,29,37)(16,48)(17,50)(18,51)(19,24)(20,26)(23,56)(27,47)(30,46)(32,61,45,63)(33,62,42,60)(34,57,52,39)(36,58,59,41)(49,54)(53,55)Polycyclic presentation:
2^8|a^2=eg,b^2=h,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fg,g^b=gh,e^c=eh,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13491, #499369, #499604, … |
| maximal normal subgroup of | C2×#13491, #499369, #499604, … |
| maximal quotient of | #146049, #146056, #146069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6479);