#414253

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

935 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321449);