#466144

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373340);