C₁₂₇×#12889 #44576562

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

Aliases

direct product

C₁₂₇×#12889

Subgroups

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

310 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 171
transitive degree	μ_t(G) = 32512

A minimal faithful permutation representation — 3 generators on 171 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,27)(16,21)(17,35)(18,38)(19,39)(20,26)(22,31)(23,28)(24,37)(25,36)(29,41)(30,40)(32,44)(33,43)(34,42)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(6,9)(10,12)(13,15,28,33,19,30,37,34,20,31,18,29,36,32,17,16)(14,21,35,44,25,41,38,22,26,42,24,40,39,43,23,27);(5,7,8,11)(6,10,9,12)(13,16,17,32,36,29,18,31,20,34,37,30,19,33,28,15)(14,22,23,41,39,44,24,21,26,27,38,43,25,40,35,42)

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, 5877);