C₁₂₇×#13333 #44577006

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#13333

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂×#135×C₁₂₇	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

546 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 163
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 163 points:

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

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6321);