#153576

order	512
nilpotency class	6
isoclinism	isoclinic class of #13742
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
socle	C₂	C₂×#1423
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 ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=df,d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,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, 60772);