#200724

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	#60700	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 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)(3,12,13,16)(4,8,14,15)(5,9)(6,10)(7,11)(17,18)(19,23,28,32)(20,25)(21,24)(22,26)(27,30,29,31);(1,3,6,15,7,13,5,8)(2,4,10,16,11,14,9,12)(17,19)(18,23,26,32)(20,29)(21,27)(22,28)(24,30,25,31);(1,4)(2,8)(3,10)(5,12)(6,16)(7,14)(9,13)(11,15)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(1,5,7,6)(2,9,11,10)(3,8,13,15)(4,12,14,16)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,30,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,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, 107920);