#443543

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=g,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 350739);