#6440843

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,e^2=i,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,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, 6348039);