#415778

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7)(3,8,4,6)(9,10,16,20,15,21,14,32)(11,22,25,17,26,37,33,35)(12,27,24,38,29,18,28,39)(13,19,31,36,30,34,23,40);(1,3,5,4)(2,6,7,8)(9,11,14,33,15,26,16,25)(10,17,32,22,21,35,20,37)(12,13,24,31,29,30,28,23)(18,36,39,34,27,40,38,19);(1,4,5,3)(2,6,7,8)(9,12,14,24,15,29,16,28)(10,18,32,39,21,27,20,38)(11,23,33,13,26,31,25,30)(17,34,22,40,35,19,37,36);(9,13,16,23,15,30,14,31)(10,19,20,40,21,34,32,36)(11,24,25,12,26,28,33,29)(17,18,37,38,35,27,22,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322974);