C₁₂₇×#7447 #44571120

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7447

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1361
commutator	C₂×C₁₆	C₂×C₄×C₁₂₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₂₇
prime radical	C₂×D₁₆×C₁₂₇	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

130 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 191
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 191 points:

(1,2,5,12,30,15,32,13,9,16,6,14,8,35,7,33)(3,17,19,41,36,49,20,54,22,50,34,64,21,55,31,48)(4,23,25,45,59,43,61,56,29,57,26,11,28,42,27,44)(10,24,37,53,39,62,46,52,40,58,63,18,47,51,38,60)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3)(2,10)(4,24)(5,20)(6,31)(7,34)(8,36)(9,22)(11,41)(12,46)(13,37)(14,38)(15,47)(16,40)(17,43)(18,27)(19,32)(21,30)(23,49)(25,52)(26,60)(28,62)(29,58)(33,63)(35,39)(42,50)(44,48)(45,64)(51,59)(53,61)(54,56)(55,57)

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, 435);