C₁₂₇×#12897 #44576570

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#12897

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#117
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#432
prime radical	C₂³×C₁₂₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#97×C₁₂₇ ◁ G

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

normal subgroups only

This group has 278 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) = 167
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 167 points:

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

Conjugacy classes & character table

8128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5885);