C₁₂₇×#8505 #44572178

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#8505

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#120
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#379
prime radical	C₁₂₇×#3531	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

648 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 175
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 175 points:

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

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, 1493);