#357208

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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)(7,17,22,18)(8,10)(9,14)(11,15)(12,16)(13,20,29,21)(19,30,35,33)(23,25)(24,36,34,28)(26,31)(27,32);(5,7,11,20,12,22,10,21)(6,13,15,17,16,29,8,18)(9,24,26,35,27,34,25,19)(14,30,31,36,32,33,23,28);(1,2,4,3)(5,8,10,16,12,15,11,6)(7,18,21,29,22,17,20,13)(9,23,25,32,27,31,26,14)(19,33,34,36,35,30,24,28);(1,3,4,2)(5,9,11,26,12,27,10,25)(6,14,15,31,16,32,8,23)(7,19,21,34,22,35,20,24)(13,28,18,33,29,36,17,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264404);