#267089

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174285);