#94693

order	512
nilpotency class	6
isoclinism	isoclinic class of #94690
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₂²×C₃₂	C₂²
prime radical	C₂³	SD₆₄
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 136 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,21,18,25,17,55)(11,26,28,78,33,81,32,93)(12,34,37,94,42,65,41,111)(14,46,43,112,15,50,44,114)(16,52,45,77,54,73,48,24)(19,56,57,27,61,82,68,85)(20,62,64,39,67,104,118,38)(22,71,69,36,23,74,70,49)(29,89,86,135,30,90,87,129)(31,91,88,134,92,127,53,80)(35,72,98,132,83,75,102,131)(40,108,105,117,110,136,106,97)(47,79,115,128,51,63,113,126)(58,121,119,116,59,109,120,101)(60,123,76,100,124,84,122,99)(66,107,95,125,130,103,96,133);(1,3)(2,5)(4,8)(6,7)(9,11,18,33)(10,19,25,61)(12,35,42,83)(13,32,17,28)(14,47,15,51)(16,53,54,88)(20,63,67,79)(21,68,55,57)(22,72,23,75)(24,76,77,122)(26,34,81,65)(27,38,85,39)(29,36,30,49)(31,45,92,48)(37,102,41,98)(40,109,110,121)(43,113,44,115)(46,58,50,59)(52,60,73,124)(56,62,82,104)(64,128,118,126)(66,129,130,135)(69,131,70,132)(71,87,74,86)(78,111,93,94)(80,108,134,136)(84,133,123,125)(89,95,90,96)(91,97,127,117)(99,103,100,107)(101,105,116,106)(112,120,114,119)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1889);