#452627

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

645 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)(9,10,15,30,14,19,20,18)(11,16,31,33,23,34,24,37)(12,25,27,17,26,38,13,28)(21,35,29,39,36,40,22,32);(1,3)(2,5)(4,7)(6,8)(9,11,20,31,14,23,15,24)(10,16,18,33,19,34,30,37)(12,21,13,29,26,36,27,22)(17,32,25,35,28,39,38,40);(3,7)(5,8)(9,12,15,27,14,26,20,13)(10,17,18,25,19,28,30,38)(11,21,24,22,23,36,31,29)(16,32,33,35,34,39,37,40);(9,13,20,26,14,27,15,12)(10,17,18,25,19,28,30,38)(11,22,31,21,23,29,24,36)(16,32,33,35,34,39,37,40)

Polycyclic presentation:

2^9|a^2=efgh,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, 359823);