C₁₂₇×#12085 #44575758

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#12085

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle, prime radical	C₂²×C₁₂₇	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₈²×C₁₂₇ ◁ G

206 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 167
transitive degree	μ_t(G) = 32512

A minimal faithful permutation representation — 3 generators on 167 points:

(1,2,5,3,4,6,8,7)(9,10)(11,19)(12,23)(13,28)(14,16)(15,20)(17,26)(18,25)(21,39)(22,36)(24,34)(27,35)(29,33)(30,38)(31,40)(32,37)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,6)(3,7)(9,11,14,22)(10,15,16,30)(12,24,27,40)(13,29,25,21)(17,33,32,39)(18,34,28,31)(19,35,36,23)(20,37,38,26);(1,3,5,6,4,7,8,2)(10,16)(15,30)(17,32)(18,28)(19,36)(23,35)(31,34)(33,39)

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, 5073);