C₁₂₇×#13435 #44577108

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

Aliases

direct product

C₁₂₇×#13435

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₁₂₇
prime radical	C₂×#41×C₁₂₇	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#41×C₁₂₇ ◁ G

408 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 86 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:

(1,2,7,12)(3,13,17,35)(4,18,22,47)(5,23,26,10)(6,27,28,11)(8,24,31,15)(9,32,34,21)(14,39,19,48)(16,43,44,38)(20,51,53,46)(25,36,57,55)(29,59,61,64)(30,42,58,56)(33,52,54,62)(37,50,63,41)(40,60,49,45)(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);(1,3,7,17)(2,8,12,31)(4,19,22,14)(5,24,26,15)(6,16,28,44)(9,29,34,61)(10,35,23,13)(11,30,27,58)(18,45,47,60)(20,40,53,49)(21,41,32,50)(25,56,57,42)(33,63,54,37)(36,43,55,38)(39,51,48,46)(52,59,62,64);(1,4,5,20,7,22,26,53)(2,9,10,33,12,34,23,54)(3,14,15,40,17,19,24,49)(6,21,25,52,28,32,57,62)(8,29,13,37,31,61,35,63)(11,18,36,46,27,47,55,51)(16,41,42,64,44,50,56,59)(30,60,38,39,58,45,43,48)

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6423);