C₁₂₇×#9502 #44573175

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₂₇×#9502

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#386
commutator	C₂⁴	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂⁵	C₂³×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂⁵×C₁₂₇ ◁ G

660 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 4064

A minimal faithful permutation representation — 3 generators on 159 points:

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

Conjugacy classes & character table

4699 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 2490);