#200987

order	512
nilpotency class	3
isoclinism	isoclinic class of #200920
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

1049 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108183);