C₁₂₇×#13710 #44577383

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13710

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
prime radical	#508×C₁₂₇	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

160 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 159
transitive degree	μ_t(G) = 4064

A minimal faithful permutation representation — 3 generators on 159 points:

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

Conjugacy classes & character table

4699 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6698);