#411480

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 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)(3,8)(4,6)(5,7)(9,10)(11,21,24,30)(12,17)(13,19)(14,18)(15,20)(16,23,31,25)(22,37,39,36)(26,34)(27,33)(28,35)(29,40,32,38);(1,3,5,4)(2,6,7,8)(9,11)(10,16,20,31)(12,22)(13,23)(14,25)(15,24)(17,32,35,29)(18,30,19,21)(26,38)(27,40)(28,39)(33,36,34,37);(1,4,5,3)(2,6,7,8)(9,12,13,26,15,28,14,27)(10,17,19,34,20,35,18,33)(11,22,23,38,24,39,25,40)(16,29,21,36,31,32,30,37);(10,18,20,19)(16,30,31,21)(17,33,35,34)(29,37,32,36)

Polycyclic presentation:

2^9|a^2=f,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, 318676);