C₁₂₇×#7040 #44570713

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#7040

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#95
commutator	C₂²×C₄	C₄²×C₁₂₇
Frattini, cosocle	C₄³	C₂²×C₁₂₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

292 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 159
transitive degree	μ_t(G) = 32512

A minimal faithful permutation representation — 2 generators on 159 points:

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

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, 28);