#420273

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 327469);