C₁₂₇×#7204 #44570877

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₂₇×#7204

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#99
commutator	C₂³	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×D₄	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#342
prime radical	C₂²×D₄×C₁₂₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×D₄×C₁₂₇ ◁ G

380 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 143
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 143 points:

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

Conjugacy classes & character table

6604 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 192);