#474440

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
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

527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381636);