C₁₂₇×#13438 #44577111

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

Aliases

direct product

C₁₂₇×#13438

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

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

normal subgroups only

This group has 406 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:

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

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, 6426);