C₁₂₇×#12088 #44575761

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#12088

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	#442
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ #322×C₁₂₇ ◁ G

198 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 167
transitive degree	μ_t(G) = 32512

A minimal faithful permutation representation — 3 generators on 167 points:

(1,2,5,3,4,6,8,7)(9,10,13,17,15,19,14,30)(11,20,23,38,21,39,24,35)(12,25,28,37,27,36,29,18)(16,31,32,22,33,26,34,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,6)(3,7)(9,11,15,21)(10,16,19,33)(12,26,27,31)(13,24,14,23)(17,34,30,32)(18,35,37,38)(20,25,39,36)(22,29,40,28);(1,3,5,6,4,7,8,2)(10,17,19,30)(11,21)(12,27)(16,32,33,34)(18,36,37,25)(20,35,39,38)(23,24)(28,29)

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, 5076);