#414254

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,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, 321450);