C₁₂₇×#13502 #44577175

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

Aliases

direct product

C₁₂₇×#13502

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2077
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6490);