#6470818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470818
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

1980 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)(3,5)(4,6)(7,8)(10,15)(11,19)(13,24)(16,28)(17,26)(18,27)(20,32)(23,35)(25,30)(29,38)(31,37)(36,39);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,18,24,27)(11,16,21,25,19,30,33,28)(12,17,23,26,22,31,35,37)(20,29,34,36,32,39,40,38);(3,7)(5,8)(9,11,13,21,14,19,24,33)(10,16,15,25,18,30,27,28)(12,20,23,34,22,32,35,40)(17,29,26,36,31,39,37,38);(3,7)(5,8)(9,12)(10,17)(11,20)(13,23)(14,22)(15,26)(16,29)(18,31)(19,32)(21,34)(24,35)(25,36)(27,37)(28,38)(30,39)(33,40);(2,6)(5,8)(12,22)(17,31)(20,32)(23,35)(26,37)(29,39)(34,40)(36,38)

Polycyclic presentation:

2^9|b^2=f,c^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6378014);