#200723

order	512
nilpotency class	3
isoclinism	isoclinic class of #200710
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 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)(3,5)(4,12,15,10)(6,11)(7,9)(8,13,16,14)(17,18)(19,26,27,30)(20,23,28,29)(21,24)(22,25)(31,32);(1,3,5,2,7,11,6,9)(4,10,14,8,15,12,13,16)(17,19,21,29,22,27,31,23)(18,20,24,30,25,28,32,26);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20)(18,23)(19,24)(21,30)(22,28)(25,29)(26,31)(27,32);(1,5,7,6)(2,3,9,11)(4,13,15,14)(8,12,16,10)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107919);