#466708

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

780 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)(3,6,7,8)(9,10,14,19)(11,20,22,17)(12,15,27,16)(13,28,30,18)(21,33,36,34)(23,25,24,35)(26,32,29,31)(37,40,38,39);(2,5)(6,8)(9,11,16,24,14,22,15,23)(10,17,12,25,19,20,27,35)(13,21,32,38,30,36,31,37)(18,33,26,39,28,34,29,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,19,14,27,16,10)(11,17,23,25,22,20,24,35)(13,29,31,18,30,26,32,28)(21,34,37,40,36,33,38,39);(3,7)(6,8)(9,13,15,31,14,30,16,32)(10,18,12,26,19,28,27,29)(11,21,23,37,22,36,24,38)(17,33,25,39,20,34,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373904);