#200974

order	512
nilpotency class	3
isoclinism	isoclinic class of #200914
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	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1026 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:

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

Polycyclic presentation:

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

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, 108170);