C₁₂₇×#13712 #44577385

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13712

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
prime radical	C₁₂₇×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂²×C₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 186 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) = 167
transitive degree	μ_t(G) = 16256

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

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,26)(14,27)(17,23)(18,24)(19,22)(20,28)(21,38)(25,34)(29,35)(30,36)(31,39)(32,37)(33,40)(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);(2,5)(6,8)(10,15)(16,28)(17,29)(18,30)(19,27)(26,38)(31,33)(32,34);(1,3,4,7)(2,6,5,8)(10,16)(12,23)(13,24)(14,22)(15,28)(17,18)(19,34)(20,35)(21,36)(25,40)(26,31)(27,32)(29,30)(33,38)(37,39)

Conjugacy classes & character table

6223 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6700);