C₁₂₇×#10586 #44574259

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#10586

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

350 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

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, 3574);