C₁₂₇×#13667 #44577340

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

Aliases

direct product

C₁₂₇×#13667

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×M₃₂×C₁₂₇ ◁ G

294 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Conjugacy classes & character table

3937 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6655);