#410407

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6,8,12,10,14,9,13)(7,15,16,20,17,11,18,19);(2,4)(5,7,9,16,10,17,8,18)(6,11,13,19,14,15,12,20);(1,2,3,4)(6,12,14,13)(11,19,15,20);(6,13,14,12)(11,19,15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216230, #72321741
maximal normal subgroup of	#72216230, #72321741
maximal quotient of	#72215689

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317603);