C₁₂₇×#13485 #44577158

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13485

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂×#135×C₁₂₇	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

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

normal subgroups only

This group has 314 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:

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

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, 6473);