C₁₂₇×#8794 #44572467

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₁₂₇×#8794

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#382
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 167
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 167 points:

(1,2,6,3)(4,5,8,7)(9,10)(11,19,22,15)(12,23,26,38)(13,18)(14,17)(16,25,31,35)(20,34)(21,33,37,29)(24,39)(27,32)(28,36)(30,40)(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);(1,3,5,8,6,2,7,4)(9,11,13,21,14,22,27,37)(10,15,18,29,17,19,32,33)(12,24,25,40,26,20,35,36)(16,30,23,34,31,28,38,39);(1,4,5,3,6,8,7,2)(9,12)(10,16,17,31)(11,20)(13,25)(14,26)(15,28,19,30)(18,23,32,38)(21,36)(22,24)(27,35)(29,39,33,34)(37,40)

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 1782);