C₁₂₇×#10527 #44574200

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#10527

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
prime radical	#532×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

348 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 159
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 159 points:

(1,2,5,9)(3,10,12,7)(4,6,13,8)(11,15,16,14)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31)(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);(2,6)(3,11)(4,13)(7,10)(8,9)(12,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,27)(26,31);(1,3,4,11,5,12,13,16)(2,7,8,15,9,10,6,14)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32)

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 3515);