#118936

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,11,16)(7,17,18,20)(8,12,19,13)(9,15,10,14)(21,22,25,27,26,30,24,29)(23,31,33,35,32,28,34,36);(1,2)(3,4)(5,7)(6,12)(8,10)(9,19)(11,18)(13,16)(14,20)(15,17)(22,27)(23,32)(24,25)(28,36)(29,30)(31,35);(2,4)(5,8,9,18,11,19,10,7)(6,13,14,20,16,12,15,17)(21,23,24,33,26,32,25,34)(22,28,29,36,30,31,27,35)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26132);