C₂×C₁₂₇×#1415 #44577286

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₁₂₇×#1415

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₂₇	D₁₆
commutator	C₁₆	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
prime radical	C₂²×D₁₆×C₁₂₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂²×C₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 342 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) = 165
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 165 points:

(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165);(3,4,5,6)(7,8)(9,13)(10,22)(11,25)(12,17)(14,18)(15,19)(16,23)(20,32)(21,28)(24,38)(26,34)(27,33)(29,37)(30,35)(31,36);(8,13)(9,18)(10,19)(11,23)(14,15)(16,32)(17,28)(20,37)(21,35)(22,26)(24,36)(25,27)(29,38)(30,31)(33,34)

Conjugacy classes & character table

9652 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6601);