C₁₂₇×#13684 #44577357

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#13684

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₂₇
prime radical	C₁₂₇×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄⋊Q₈×C₁₂₇ ◁ G

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 159
transitive degree	μ_t(G) = 4064

A minimal faithful permutation representation — 3 generators on 159 points:

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

Conjugacy classes & character table

3175 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6672);