C₁₂₇×#13573 #44577246

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#13573

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₂₇
prime radical	C₂×2₋¹⁺⁴×C₁₂₇	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄⋊Q₈×C₁₂₇ ◁ G

470 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 90 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:

(1,2,4,9,6,11,19,29)(3,12,15,23,13,25,31,7)(5,10,21,8,22,26,18,27)(14,30,20,28,17,24,16,32)(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)(2,7)(4,17)(5,20)(6,13)(8,25)(9,28)(10,30)(11,23)(12,27)(14,19)(15,18)(16,22)(21,31)(24,26)(29,32);(3,13)(7,23)(12,25)(14,17)(15,31)(16,20)(24,30)(28,32)

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, 6561);