#470705

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
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

637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377901);