C₁₂₇×#7081 #44570754

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7081

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#327
commutator	C₂²×C₈	C₂×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₁₂₇
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄⋊C₈×C₁₂₇ ◁ G

172 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 8128

A minimal faithful permutation representation — 2 generators on 159 points:

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

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, 69);