Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | Aut(G) = #72314138 (order 4096) · Inn(G) = #469 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2054 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 193 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)(3,16)(4,22)(5,29)(6,32)(7,12)(8,34)(9,15)(10,18)(11,25)(13,31)(14,30)(17,36)(19,46)(20,38)(21,45)(23,47)(24,55)(26,44)(27,58)(28,52)(33,40)(35,37)(39,51)(41,53)(42,56)(43,57)(48,60)(49,63)(50,62)(54,64)(59,61);(1,3)(2,10)(4,23)(5,19)(6,33)(7,18)(8,35)(9,21)(11,41)(12,37)(13,45)(14,46)(15,40)(16,29)(17,26)(20,30)(22,36)(24,49)(25,50)(27,48)(28,54)(31,51)(32,38)(34,39)(42,63)(43,64)(44,60)(47,55)(52,62)(53,56)(57,59)(58,61);(1,4,5,24,30,56,6,25,9,28,31,57,8,27,7,26)(2,11,12,42,34,55,13,22,15,44,32,58,14,43,29,52)(3,17,18,48,35,59,51,54,21,50,33,53,20,49,19,23)(10,36,16,47,46,63,38,41,40,62,45,64,39,61,37,60)Polycyclic presentation:
2^8|c^2=d,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93810, #93811, #151294, … |
| maximal normal subgroup of | #93810, #93811, #151294, … |
| maximal quotient of | #145363, #145364, #145389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6435);