#449561

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2009 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356757);