#119321

order	512
nilpotency class	3
isoclinism	isoclinic class of #100227
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

574 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,f^2=i,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=fi,g^b=ghi,e^c=eh,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, 26517);