#415557

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

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

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, 322753);