C₁₂₇×#13676 #44577349

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

Aliases

direct product

C₁₂₇×#13676

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2196
commutator, Frattini, cosocle	#127	C₂³×C₁₂₇
prime radical	C₁₂₇×#3291	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ #127×C₁₂₇ ◁ G

394 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

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, 6664);