#470853

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

724 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)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,30)(11,20,24,32,25,21,23,16)(12,26,28,33,29,40,27,17)(22,36,38,35,39,31,37,34);(2,5)(6,8)(9,11,13,23,15,25,14,24)(10,16,18,20,19,32,30,21)(12,22,27,37,29,39,28,38)(17,31,26,34,33,36,40,35);(2,5)(3,7)(9,10,15,19)(11,21)(12,26,29,40)(13,30,14,18)(16,24)(17,28,33,27)(20,25)(22,31)(23,32)(34,38)(35,37)(36,39);(1,3,4,7)(2,6,5,8)(9,12,13,27,15,29,14,28)(10,17,18,26,19,33,30,40)(11,22,23,37,25,39,24,38)(16,31,20,34,32,36,21,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=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, 378049);