#114444

order	512
nilpotency class	3
isoclinism	isoclinic class of #114437
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

852 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)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18,22,26,23,19,21,25)(20,24,30,28,31,27,29,32);(1,3,6,12)(2,7,9,16)(4,13,5,11)(8,15,14,10)(17,19,23,18)(20,28,31,32)(21,25,22,26)(24,29,27,30);(3,11)(4,5)(7,15)(8,14)(10,16)(12,13)(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,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,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, 21640);