C₁₂₇×#13450 #44577123

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

Aliases

direct product

C₁₂₇×#13450

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

410 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,4)(9,10,17,24)(11,25,30,40)(12,31,28,38)(13,21,32,22)(14,20,15,35)(16,36,26,23)(18,34,39,27)(19,33,37,29)(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);(5,6)(7,8)(9,11,17,30)(10,18,24,39)(12,32,28,13)(14,27,15,34)(16,33,26,29)(19,20,37,35)(21,25,22,40)(23,31,36,38);(2,4)(6,8)(9,12,13,11,26,27,15,33,17,28,32,30,16,34,14,29)(10,19,20,39,36,40,22,38,24,37,35,18,23,25,21,31)

Conjugacy classes & character table

6604 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6438);