C₁₂₇×#7129 #44570802

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₂₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₂₇×#7129

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#329
commutator	C₄²	C₂×C₈×C₁₂₇
Frattini, cosocle	#376	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1272
prime radical	C₂³×C₁₂₇	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #376×C₁₂₇ ◁ G

210 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,11,13,26,14,27,19,30)(4,15,17,31)(5,9)(6,10)(7,12,21,28,23,16,22,29)(8,18,24,32)(20,25)(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);(1,3,6,14)(2,7)(4,16,17,12)(5,19,20,13)(8,11)(9,21)(10,23)(15,30)(18,29,32,28)(22,25)(24,27)(26,31)

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 117);