C₁₂₇×#13330 #44577003

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

Aliases

direct product

C₁₂₇×#13330

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

558 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,10,15,20)(11,21,12,24)(13,27,32,40)(14,33,30,19)(16,25,17,22)(18,36,39,31)(23,38,26,37)(28,35,29,34)(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);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,20,17)(13,28,32,29)(14,26,30,23)(18,37,39,38)(19,35,33,34)(21,40,24,27)(22,36,25,31);(6,8)(9,12,15,11)(10,17,20,16)(13,29,32,28)(14,23,30,26)(18,38,39,37)(19,34,33,35)(21,27,24,40)(22,31,25,36)

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, 6318);