#94739

order	512
nilpotency class	6
isoclinism	isoclinic class of #94737
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7444
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
prime radical	C₂²	#1419
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

80 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fgi,d^2=efgi,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=defg,e^b=eg,f^b=fgi,g^b=gh,h^b=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1935);