#6448768

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	#62845	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3215 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)(11,19)(12,16)(13,25)(14,18)(15,21)(17,28)(20,36)(22,34)(23,32)(24,35)(26,30)(27,33)(29,39)(31,38)(37,40);(9,11,14,22)(10,15,18,30)(12,20,23,37)(13,26,28,21)(16,31,32,29)(17,34,25,19)(24,39,27,38)(33,40,35,36);(2,5)(6,8)(9,12)(10,16)(11,20)(13,27)(14,23)(15,29)(17,35)(18,32)(19,36)(21,39)(22,37)(24,28)(25,33)(26,38)(30,31)(34,40);(1,3,4,7)(2,6,5,8)(13,28)(15,30)(17,25)(21,26)(24,27)(29,31)(33,35)(38,39);(12,23)(16,32)(20,37)(24,27)(29,31)(33,35)(36,40)(38,39)

Polycyclic presentation:

2^9|b^2=i,d^2=g,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, 6355964);