#470972

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
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

636 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)(9,10,14,21,16,22,15,20)(11,19,13,17,25,33,24,34)(12,26,30,37,29,36,28,18)(23,32,31,40,27,35,39,38);(1,2,4,5)(3,6,7,8)(9,11,15,24,16,25,14,13)(10,17,21,33,22,34,20,19)(12,27,28,31,29,23,30,39)(18,32,26,40,37,35,36,38);(3,7)(6,8)(9,12,15,28,16,29,14,30)(10,18,21,26,22,37,20,36)(11,23,24,39,25,27,13,31)(17,32,33,40,34,35,19,38);(1,3,4,7)(2,6,5,8)(9,13,14,25,16,24,15,11)(10,19,20,34,22,33,21,17)(12,27,28,31,29,23,30,39)(18,35,26,38,37,32,36,40)

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^c=de,e^c=ei,h^c=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, 378168);