#453938

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

569 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=fgh,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,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, 361134);