#416537

order	512
nilpotency class	3
isoclinism	isoclinic class of #397240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1104 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fh,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=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, 323733);