C₁₂₇×#12306 #44575979

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

Aliases

direct product

C₁₂₇×#12306

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
prime radical	#469×C₁₂₇	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×M₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 274 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:

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

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, 5294);