C₁₂₇×#12303 #44575976

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

Aliases

direct product

C₁₂₇×#12303

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
prime radical	C₁₂₇×#3204	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×M₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 322 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:

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

Conjugacy classes & character table

3175 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5291);