#470973

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

626 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=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, 378169);