#94692

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	#13694	C₂
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

93 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,20,17,24,16,42)(11,25,28,51,32,18,31,43)(13,36,33,68,14,39,34,69)(15,40,35,58,41,55,38,23)(19,47,48,26,50,60,70,29)(21,54,52,27,22,56,53,37)(30,65,61,46,67,57,62,59)(44,72,49,64,45,66,71,63);(1,3)(2,5)(4,8)(6,7)(10,18)(11,26)(12,16)(13,37)(14,27)(15,38)(19,22)(20,51)(21,50)(23,57)(24,25)(28,47)(29,32)(30,66)(31,60)(33,56)(34,54)(35,41)(36,44)(39,45)(40,46)(42,43)(48,52)(49,69)(53,70)(55,59)(58,65)(61,64)(62,63)(67,72)(68,71)

Polycyclic presentation:

2^9|a^2=d,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, 1888);