#357209

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
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

1288 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)(9,24)(11,15)(12,16)(13,20,30,21)(14,28)(19,33,35,31)(23,27)(25,29,34,36)(26,32);(5,7,11,20,12,22,10,21)(6,13,15,17,16,30,8,18)(9,25,27,35,28,34,26,19)(14,31,32,36,24,33,23,29);(1,2,4,3)(5,8,10,16,12,15,11,6)(7,18,21,30,22,17,20,13)(9,23,26,24,28,32,27,14)(19,33,34,36,35,31,25,29);(1,3,4,2)(5,9,11,27,12,28,10,26)(6,14,15,32,16,24,8,23)(7,19,21,34,22,35,20,25)(13,29,18,33,30,36,17,31)

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,d^a=di,f^a=fi,d^b=de,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264405);