C₁₂₇×#7161 #44570834

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#7161

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#106
commutator	C₈	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#334
prime radical	C₂³×C₁₂₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

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

normal subgroups only

This group has 202 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) = 163
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 163 points:

(1,2)(3,4)(5,6,9,15,11,16,26,32,12,17,25,14,10,28,8,24)(7,18,20,29,22,33,19,30,23,31,27,36,21,35,34,13)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(2,4)(5,7,8,19,10,21,25,20,12,23,26,34,11,22,9,27)(6,13,14,18,16,29,24,33,17,30,15,31,28,36,32,35)

Conjugacy classes & character table

8128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 149);