#6440361

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62848	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4109 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)(9,10)(11,16)(12,17)(13,15)(14,19)(18,25)(20,31)(21,32)(22,30)(23,33)(24,34)(26,28)(27,29)(35,40)(36,38)(37,39);(1,3,4,7)(2,5,6,8)(10,15)(13,25)(16,30)(17,33)(18,19)(22,26)(23,27)(28,32)(29,34)(31,38)(36,37)(39,40);(2,6)(5,8)(9,11)(10,16)(12,20)(13,26)(14,21)(15,28)(17,31)(18,30)(19,32)(22,25)(23,37)(24,35)(27,36)(29,38)(33,39)(34,40);(9,12,14,24)(10,17,19,34)(11,20,21,35)(13,27,25,23)(15,29,18,33)(16,31,32,40)(22,37,26,36)(28,38,30,39);(11,21)(16,32)(20,35)(22,26)(28,30)(31,40)(36,37)(38,39)

Polycyclic presentation:

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

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, 6347557);