#390643

order	512
nilpotency class	3
isoclinism	isoclinic class of #219542
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ghi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=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, 297839);