#452534

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

905 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)(3,5,7,8)(9,10,15,19,14,20,13,16)(11,21,24,32,22,17,29,30)(12,25,28,34,27,18,26,31)(23,35,38,39,36,33,37,40);(1,3)(2,5)(4,7)(6,8)(10,16,20,19)(11,22)(12,26,27,28)(17,32,21,30)(18,25)(23,37,36,38)(24,29)(31,34);(3,7)(5,8)(9,11,13,24,14,22,15,29)(10,17,16,30,20,21,19,32)(12,23,28,37,27,36,26,38)(18,33,31,39,25,35,34,40);(9,12,13,28,14,27,15,26)(10,18,16,31,20,25,19,34)(11,23,24,37,22,36,29,38)(17,33,30,39,21,35,32,40)

Polycyclic presentation:

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

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, 359730);