#475717

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

739 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=egh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 382913);