C₁₂₇×#10632 #44574305

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

Aliases

direct product

C₁₂₇×#10632

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

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

normal subgroups only

This group has 310 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) = 167
transitive degree	μ_t(G) = 8128

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

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

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, 3620);