#390797

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
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

835 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=efgh,d^2=ei,e^2=i,f^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,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, 297993);