#411494

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

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

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, 318690);