C₁₂₇×#7309 #44570982

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#7309

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#342
commutator	C₂²×C₄	C₄²×C₁₂₇
Frattini, cosocle	C₂²×#41	C₂²×C₁₂₇
prime radical	C₂²×#41×C₁₂₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×#41×C₁₂₇ ◁ G

470 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 167
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 167 points:

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

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 297);