#453895

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

576 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)(3,7,8,5)(9,10,13,17,15,20,14,19)(11,21,23,33,24,32,30,16)(12,25,27,36,29,35,28,18)(22,31,26,34,37,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,23,15,24,13,30)(10,16,19,21,20,33,17,32)(12,26,28,37,29,38,27,22)(18,34,35,40,36,39,25,31);(10,17,20,19)(16,21,33,32)(18,25,36,35)(31,34,40,39);(3,8)(5,7)(9,12,13,27,15,29,14,28)(10,18,19,35,20,36,17,25)(11,22,23,26,24,37,30,38)(16,31,32,39,33,40,21,34)

Polycyclic presentation:

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

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, 361091);