C₁₂₇×#8327 #44572000

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

Aliases

direct product

C₁₂₇×#8327

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

684 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 158 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:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,36)(28,42,44,41)(30,40)(32,39,48,46)(35,38)(37,43,45,47)(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);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,30)(18,23,26,38)(20,28,33,44)(21,35,36,29)(24,37,31,45)(25,40,34,27)(32,47,48,43)(39,42,46,41);(3,9)(6,12)(10,15)(13,16)(17,20,22,33)(18,24,26,31)(19,28,30,44)(21,32,36,48)(23,37,38,45)(25,39,34,46)(27,41,40,42)(29,43,35,47)

Conjugacy classes & character table

5842 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 1315);