#414255

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(9,10,16,19,15,20,14,30)(11,21,32,34,25,38,24,17)(12,18,31,29,28,33,27,13)(22,36,40,23,37,35,26,39);(1,3,4,8)(2,5,6,7)(9,11,14,24,15,25,16,32)(10,17,19,21,20,34,30,38)(12,26,27,37,28,40,31,22)(13,23,18,35,29,39,33,36);(9,12)(10,18)(11,22)(13,19)(14,31)(15,28)(16,27)(17,23)(20,33)(21,36)(24,40)(25,37)(26,32)(29,30)(34,39)(35,38);(9,13,16,33,15,29,14,18)(10,12,19,27,20,28,30,31)(11,23,32,36,25,39,24,35)(17,26,21,37,34,40,38,22)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321451);