C₁₂₇×#10339 #44574012

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

Aliases

direct product

C₁₂₇×#10339

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#392
commutator, Frattini, cosocle	C₂⁵	C₂³×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂⁵×C₁₂₇ ◁ G

738 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 4064

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

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

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 3327);