#452489

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
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

746 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)(11,16,24,34)(12,26)(13,18)(14,20)(15,30)(17,28)(19,21)(22,37,38,40)(23,31,29,32)(25,39,36,33)(27,35);(1,3)(2,5)(4,7)(6,8)(9,11,21,36,14,24,15,25)(10,16,19,33,20,34,30,39)(12,22,13,29,27,38,28,23)(17,31,26,37,18,32,35,40);(3,7)(5,8)(9,12,15,28,14,27,21,13)(10,17,19,26,20,18,30,35)(11,22,25,23,24,38,36,29)(16,31,33,37,34,32,39,40);(9,13,21,27,14,28,15,12)(10,18,19,35,20,17,30,26)(11,23,36,22,24,29,25,38)(16,32,33,40,34,31,39,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=ef,c^2=f,d^2=ef,f^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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359685);