#185381

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2246 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)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18,21,25,22,26,20,24)(19,27,28,23,30,31,29,32);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,4,7,2)(3,11,13,8)(5,14,6,9)(10,15,16,12);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92577);