#415556

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
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

692 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)(11,22,26,34)(12,18)(13,29)(14,20)(15,31)(16,21)(17,25,33,30)(19,28)(23,38,39,37)(24,32,40,35)(27,36);(1,3,5,4)(2,6,7,8)(9,11,14,30,16,26,15,25)(10,17,20,34,21,33,31,22)(12,23,27,40,28,39,13,24)(18,35,36,37,19,32,29,38);(1,4,5,3)(2,6,7,8)(9,12,14,27,16,28,15,13)(10,18,20,36,21,19,31,29)(11,23,30,40,26,39,25,24)(17,32,34,38,33,35,22,37);(9,13,15,28,16,27,14,12)(10,19,20,29,21,18,31,36)(11,24,25,39,26,40,30,23)(17,32,34,38,33,35,22,37)

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322752);