C₁₂₇×#12593 #44576266

order	32512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₁₂₇ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₁₂₇×#12593

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#97
prime radical	C₂³×C₁₂₇	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ G

338 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 3 generators on 167 points:

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

Conjugacy classes & character table

11176 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5581);