C₁₂₇×#7452 #44571125

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₂₇×#7452

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	#357
commutator	C₂×C₁₆	C₂×C₄×C₁₂₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

102 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 32512

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

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

Conjugacy classes & character table

6223 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 440);