C₁₂₇×#7058 #44570731

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#7058

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#99
commutator	C₂²×C₄	C₂×C₈×C₁₂₇
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#323
prime radical	C₂²×#41×C₁₂₇	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₂²⋊C₈×C₁₂₇ ◁ G

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

normal subgroups only

This group has 354 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) = 8128

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

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

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, 46);