#409296

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
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	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

945 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)(9,10,15,28,13,16,14,19)(11,20,27,32,23,17,21,30)(12,24,29,33,25,18,26,37)(22,34,38,40,36,31,35,39);(1,3,5,4)(2,6,7,8)(10,16)(11,21,23,27)(17,30,20,32)(18,24)(19,28)(22,35,36,38)(31,39,34,40)(33,37);(1,4,5,3)(2,6,7,8)(9,11,13,23)(10,17,16,20)(12,22,25,36)(14,27,15,21)(18,31,24,34)(19,30,28,32)(26,38,29,35)(33,39,37,40);(9,12)(10,18)(11,22)(13,25)(14,26)(15,29)(16,24)(17,31)(19,33)(20,34)(21,35)(23,36)(27,38)(28,37)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,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, 316492);