#149389

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1920
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56585);