C₁₂₇×#7246 #44570919

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₂₇×#7246

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#340
commutator	C₂×C₈	C₄²×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
prime radical	C₂³×C₁₂₇	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

184 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 16256

A minimal faithful permutation representation — 2 generators on 167 points:

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

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, 234);