#186586

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62800	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,9,6,7,4,14)(3,10,13,8,12,16,11,15)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(2,7)(4,5)(8,15)(11,13)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,3)(2,8)(4,11)(5,13)(6,12)(7,15)(9,16)(10,14);(1,4,6,5)(2,9,7,14)(3,11,12,13)(8,16,15,10)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93782);