#6470426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2680 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)(3,5)(4,6)(7,8)(10,16)(11,21)(13,14)(18,31)(19,20)(23,35)(24,25)(26,34)(27,29)(30,33)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10,14,16,15,20,13,19)(11,22,24,33,25,17,21,30)(12,26,27,34,28,18,29,31)(23,32,37,39,38,36,35,40);(3,7)(5,8)(9,11)(10,17)(12,23)(13,21)(14,24)(15,25)(16,30)(18,32)(19,33)(20,22)(26,36)(27,37)(28,38)(29,35)(31,39)(34,40);(3,7)(5,8)(9,12,15,28)(10,18,20,26)(11,23,25,38)(13,29,14,27)(16,31,19,34)(17,32,22,36)(21,35,24,37)(30,39,33,40);(2,6)(5,8)(9,13,15,14)(10,19,20,16)(11,21,25,24)(12,27,28,29)(17,33,22,30)(18,31,26,34)(23,37,38,35)(32,39,36,40)

Polycyclic presentation:

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

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, 6377622);