#464644

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1565 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)(3,6)(4,5)(7,8)(9,10,14,19,13,18,15,20)(11,21,25,32,24,16,26,22)(12,17,29,33,28,27,30,34)(23,35,39,36,38,31,40,37);(2,5)(6,8)(9,11,15,26,13,24,14,25)(10,16,20,32,18,21,19,22)(12,23,29,39,28,38,30,40)(17,31,33,37,27,35,34,36);(1,3)(2,6)(4,7)(5,8)(9,10)(11,22)(12,27)(13,18)(14,20)(15,19)(16,25)(17,28)(21,26)(23,37)(24,32)(29,33)(30,34)(31,39)(35,40)(36,38);(3,7)(6,8)(9,12)(10,17)(11,23)(13,28)(14,30)(15,29)(16,31)(18,27)(19,34)(20,33)(21,35)(22,36)(24,38)(25,40)(26,39)(32,37)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,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, 371840);