#94835

order	512
nilpotency class	7
isoclinism	isoclinic class of #94835
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7537
commutator	C₆₄	C₂×C₄
Frattini, cosocle	C₂×C₆₄	C₂²
prime radical	#13741	C₂
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₂×C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

53 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=efi,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cei,d^b=di,e^b=ef,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230770
maximal normal subgroup of	#72230770

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2031);