C₁₂₇×#20340 #44584013

order	32512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₁₂₇×#20340

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂×C₁₂₇	C₂²×#95
prime radical	C₂²×#41×C₁₂₇	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ G

1956 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 1114 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 — 4 generators on 159 points:

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

Conjugacy classes & character table

10414 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 13328);