#201022

order	512
nilpotency class	3
isoclinism	isoclinic class of #200944
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 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)(3,9)(4,12)(8,14)(10,11)(13,16)(17,18,22,25,23,26,21,19)(20,28,31,24,30,27,29,32);(1,2,4,6,5,8,12,14)(3,7,11,13,10,15,9,16)(17,19,22,26,23,25,21,18)(20,27,29,32,30,28,31,24);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,12)(14,16)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,28)(26,32);(3,10)(7,15)(9,11)(13,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,32,28)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108218);