C₁₂₇×#13643 #44577316

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13643

Subgroups

	subgroup	G/H
center	C₈×C₁₂₇	#117
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#2163
prime radical	C₂³×C₁₂₇	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ #346×C₁₂₇ ◁ G

164 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Conjugacy classes & character table

8128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6631);