C₁₂₇×#13660 #44577333

order	32512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13660

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₂₇	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle, prime radical	C₂²×C₁₂₇	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 158 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) = 32512

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

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

Conjugacy classes & character table

14224 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6648);