#433521

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 340717);