C₁₂₇×#13342 #44577015

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#13342

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₁₂₇×#3150	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×#97×C₁₂₇ ◁ G

558 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 163
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 163 points:

(5,6,13,20)(7,15,25,35)(8,26,21,14)(9,17,29,36)(10,16,27,30)(11,19,12,18)(22,33,28,32)(23,34,24,31)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2)(3,4)(5,7)(6,14)(8,27)(9,28)(10,21)(11,24)(12,23)(13,25)(15,17)(16,34)(18,33)(19,32)(20,26)(22,29)(30,31)(35,36);(2,4)(5,8)(6,15)(7,9)(10,23)(11,22)(12,28)(13,21)(14,30)(16,26)(17,33)(18,34)(19,31)(20,35)(24,27)(25,29)(32,36)

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, 6330);