C₁₂₇×#11458 #44575131

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#11458

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	C₂×#99
commutator	C₂³	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₂×#324
prime radical	C₂⁴×C₁₂₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂²×M₁₆×C₁₂₇ ◁ G

364 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Conjugacy classes & character table

5842 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 4446);