#6440850

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348046);