#460620

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

999 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)(3,5)(4,6)(7,8)(9,10,15,19,14,18,13,20)(11,21,30,32,24,33,23,16)(12,25,27,17,29,34,28,35)(22,37,38,36,26,31,39,40);(1,3,4,7)(2,5,6,8)(9,11,14,24)(10,16)(12,26,29,22)(13,23,15,30)(17,31)(18,32)(19,21)(20,33)(25,36)(27,39,28,38)(34,40)(35,37);(2,6)(5,8)(9,12,14,29)(10,17,18,35)(11,22,24,26)(13,27,15,28)(16,31,32,37)(19,25,20,34)(21,36,33,40)(23,38,30,39);(10,18)(12,27)(13,15)(16,32)(17,34)(22,38)(23,30)(25,35)(26,39)(28,29)(31,40)(36,37)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 367816);