#475134

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	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,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, 382330);