C₁₂₇×#13648 #44577321

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13648

Subgroups

	subgroup	G/H
center	C₁₆×C₁₂₇	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle	C₂×C₁₂₇	D₄×C₁₆
prime radical	#135×C₁₂₇	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₂₇ ◁ C₂×C₃₂×C₁₂₇ ◁ G

184 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 191 points:

(1,2,6,13,7,14,31,46,8,12,32,44,34,45,37,62,9,15,33,47,35,17,49,51,5,27,29,58,30,50,55,61)(3,16,19,48,20,25,53,57,18,10,36,38,52,26,11,41,21,39,43,63,54,23,42,56,28,40,59,64,60,4,22,24)(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)(4,17)(5,28)(6,19)(7,20)(8,18)(9,21)(11,37)(12,39)(13,38)(14,26)(15,40)(16,27)(22,55)(23,45)(24,51)(25,50)(29,59)(30,60)(31,53)(32,36)(33,43)(34,52)(35,54)(41,46)(42,49)(44,63)(47,64)(48,58)(56,62)(57,61);(1,4,6,24,7,16,31,48,8,25,32,57,34,10,37,38,9,26,33,41,35,39,49,63,5,23,29,56,30,40,55,64)(2,11,13,21,14,43,46,54,12,42,44,28,45,59,62,60,15,22,47,3,17,19,51,20,27,53,58,18,50,36,61,52)

Conjugacy classes & character table

11176 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6636);