C₁₂₇×#13679 #44577352

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

Aliases

direct product

C₁₂₇×#13679

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2196
commutator, Frattini, cosocle	#124	C₂³×C₁₂₇
prime radical	C₁₂₇×#3288	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ #124×C₁₂₇ ◁ G

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

normal subgroups only

This group has 386 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:

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

Conjugacy classes & character table

2794 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6667);