#200717

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 107913);