C₁₂₇×#12676 #44576349

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₂₇×#12676

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#107
prime radical	C₂³×C₁₂₇	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 16256

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

(1,2,4,3)(5,6,8,13,21,28,35,36,10,12,23,27,9,14,22,30)(7,15,16,25,17,32,33,24,19,31,34,26,18,29,20,11)(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);(5,7,9,17,10,19,21,18)(6,11,14,25,12,24,28,26)(8,20,22,16,23,33,35,34)(13,29,30,15,27,32,36,31);(1,3,4,2)(6,12)(11,24)(13,27)(14,28)(15,31)(25,26)(29,32)(30,36)

Conjugacy classes & character table

8128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5664);