#452491

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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)(9,10,16,20,14,21,15,31)(11,17,24,33,23,19,13,30)(12,25,29,35,27,40,28,18)(22,36,38,32,37,39,26,34);(1,3)(2,5)(4,7)(6,8)(9,11,16,13,14,23,15,24)(10,17,20,30,21,19,31,33)(12,22,29,26,27,37,28,38)(18,32,25,36,35,34,40,39);(3,7)(5,8)(9,12,15,28,14,27,16,29)(10,18,20,25,21,35,31,40)(11,22,24,38,23,37,13,26)(17,32,30,36,19,34,33,39);(9,13,16,23,14,24,15,11)(10,19,20,33,21,17,31,30)(12,26,29,37,27,38,28,22)(18,34,25,39,35,32,40,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=ef,c^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,h^d=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, 359687);