#1279×C₂₅₁ #44545866

order	32128
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₂₅₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#1279×C₂₅₁

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₂₅₁	D₄
commutator	C₄	C₄×C₈×C₂₅₁
Frattini, cosocle	C₂×C₄²	C₂²×C₂₅₁
socle, prime radical	C₂³×C₂₅₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

208 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 271
transitive degree	μ_t(G) = 32128

A minimal faithful permutation representation — 2 generators on 271 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,16,19,17,20,15,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271);(2,4)(5,6,8,11,9,12,7,10)(14,18,20,19)

Conjugacy classes & character table

14056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32128, 11);