C₁₂₇×#13491 #44577164

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13491

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2073
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

272 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 191
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 191 points:

(1,2,8,29,5,11,7,13)(3,14,27,33,16,9,18,42)(4,10,21,37,22,35,28,43)(6,12,30,58,23,38,26,41)(15,44,47,62,48,31,51,60)(17,45,55,39,49,32,24,57)(19,52,50,63,53,34,54,61)(20,36,25,40,46,59,56,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,5,16)(2,9,11,14)(4,19,22,53)(6,24,23,55)(7,27,8,18)(10,34,35,52)(12,39,38,57)(13,42,29,33)(15,46,48,20)(17,30,49,26)(21,54,28,50)(25,51,56,47)(31,59,44,36)(32,58,45,41)(37,63,43,61)(40,62,64,60);(1,4)(2,10,11,35)(3,15)(5,22)(6,25)(7,28)(8,21)(9,31,14,44)(12,40,38,64)(13,43,29,37)(16,48)(17,50)(18,51)(19,24)(20,26)(23,56)(27,47)(30,46)(32,61,45,63)(33,62,42,60)(34,57,52,39)(36,58,59,41)(49,54)(53,55)

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6479);