#439025

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1443 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:

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

Polycyclic presentation:

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

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, 346221);