#94769

order	512
nilpotency class	6
isoclinism	isoclinic class of #7540
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1415
commutator	C₃₂	C₄²
Frattini, cosocle	C₂²×C₃₂	C₂²
socle	C₂	#7471
prime radical	C₂³	Q₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=i,e^2=f,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1965);