#495954

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=gi,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403150);