C₁₂₇×#12730 #44576403

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₂₇×#12730

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₂₇
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ #519×C₁₂₇ ◁ G

666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 159
transitive degree	μ_t(G) = 4064

A minimal faithful permutation representation — 3 generators on 159 points:

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

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, 5718);