#453936

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

568 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,23,26,17)(13,22,29,34)(15,19)(16,20)(18,30,27,28)(21,25,33,31)(24,35)(32,37)(36,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11,14,20,16,19,15,10)(12,17,25,21,26,23,31,33)(13,27,30,34,29,18,28,22)(24,35,39,36,38,40,37,32);(2,7)(6,8)(9,12,15,31,16,26,14,25)(10,17,19,33,20,23,11,21)(13,24,28,37,29,38,30,39)(18,32,34,40,27,36,22,35);(1,4,5,3)(2,8,7,6)(9,13,14,30,16,29,15,28)(10,18,11,22,20,27,19,34)(12,24,25,39,26,38,31,37)(17,32,21,35,23,36,33,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=egh,d^2=ef,e^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,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 361132);