C₁₂₇×#7337 #44571010

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7337

Subgroups

	subgroup	G/H
center	C₈×C₁₂₇	#99
commutator	C₂×C₄	C₂×C₁₆×C₁₂₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1314
prime radical	C₂×#41×C₁₂₇	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂×C₈×C₁₂₇ ◁ C₂×M₃₂×C₁₂₇ ◁ G

154 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 191 points:

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

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, 325);