C₁₂₇×#7312 #44570985

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#7312

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1304
commutator	C₂⁴	C₄²×C₁₂₇
Frattini, cosocle	C₂²×#41	C₂²×C₁₂₇
prime radical	C₂²×#41×C₁₂₇	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×#41×C₁₂₇ ◁ G

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

normal subgroups only

This group has 456 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) = 159
transitive degree	μ_t(G) = 4064

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

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

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 300);