Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2033 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,7,12)(3,13,18,33)(4,19,23,54)(5,25,27,10)(6,28,30,59)(8,17,34,15)(9,35,38,22)(11,40,42,26)(14,45,20,56)(16,32,51,63)(21,55,24,41)(29,39,58,37)(31,49,36,61)(43,57,64,47)(44,48,60,50)(46,62,52,53);(1,3)(2,8)(4,20)(5,17)(6,16)(7,18)(9,36)(10,33)(11,32)(12,34)(13,25)(14,23)(15,27)(19,53)(21,46)(22,57)(24,52)(26,50)(28,60)(29,61)(30,51)(31,38)(35,47)(37,43)(39,64)(40,48)(41,45)(42,63)(44,59)(49,58)(54,62)(55,56);(1,4,5,21,7,23,27,24)(2,9,10,37,12,38,25,39)(3,14,15,46,18,20,17,52)(6,29,26,35,30,58,40,22)(8,31,13,43,34,36,33,64)(11,41,28,54,42,55,59,19)(16,49,48,47,51,61,50,57)(32,56,44,62,63,45,60,53)Polycyclic presentation:
2^8|a^2=h,c^2=d,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=cf,d^b=dh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13390, #499025, #499604, … |
| maximal normal subgroup of | C2×#13390, #499025, #499604, … |
| maximal quotient of | #144687, #144779, #145003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6378);