#417291

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

817 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi,g^c=gi,h^c=hi,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, 324487);