#114478

order	512
nilpotency class	3
isoclinism	isoclinic class of #114473
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21674);