Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #482 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2092 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
147 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,5,11,30,46,6,12,8,14,32,48,7,13,31,47)(3,15,17,37,19,36,18,39,21,49,33,38,20,44,28,9)(4,22,24,10,34,41,29,50,27,45,59,43,26,57,25,42)(16,40,52,51,23,58,56,64,55,63,60,62,54,35,53,61);(1,3,8,21)(2,9,14,39)(4,23)(5,28,32,18)(6,33,31,17)(7,19,30,20)(10,40)(11,44,48,36)(12,49,47,15)(13,37,46,38)(16,34)(22,51)(24,52)(25,56)(26,55)(27,54)(29,53)(35,50)(41,61)(42,58)(43,63)(45,62)(57,64)(59,60);(1,4)(2,10)(3,16)(5,29)(6,24)(7,34)(8,27)(9,35)(11,45)(12,41)(13,50)(14,43)(15,51)(17,56)(18,52)(19,54)(20,23)(21,55)(22,48)(25,32)(26,30)(28,60)(31,59)(33,53)(36,61)(37,63)(38,40)(39,58)(42,46)(44,64)(47,57)(49,62)Polycyclic presentation:
2^8|a^2=d,b^2=de,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egh,g^b=gh,d^c=dgh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13525, #499520, #499595, … |
| maximal normal subgroup of | C2×#13525, #499520, #499595, … |
| maximal quotient of | #146186, #146187, #146188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6513);