#6446735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3985 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,23)(13,17)(14,22)(15,18)(16,26)(20,31)(21,29)(24,27)(25,36)(28,32)(30,34)(33,38)(35,39)(37,40);(2,5)(6,7)(9,11,19,10,15,22,14,18)(12,24,31,16,26,20,27,23)(13,28,32,17,30,21,29,34)(25,35,40,33,38,37,39,36);(1,3,4,8)(2,6,5,7)(9,12,15,26)(10,16,18,23)(11,20,22,24)(13,25,30,38)(14,31,19,27)(17,33,34,36)(21,35,28,37)(29,40,32,39);(9,13,15,30)(10,17,18,34)(11,21,22,28)(12,25,26,38)(14,32,19,29)(16,33,23,36)(20,35,24,37)(27,40,31,39);(12,26)(16,23)(20,24)(25,38)(27,31)(33,36)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=fg,c^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,h^b=hi,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, 6353931);