#6456437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
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
prime radical	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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)(3,6)(4,5)(7,8)(9,10,16,32,15,20,14,21)(11,22,31,17,26,40,25,35)(12,18,30,39,28,37,13,19)(23,38,24,33,27,34,29,36);(9,11)(10,17)(12,27)(13,29)(14,31)(15,26)(16,25)(18,36)(19,38)(20,35)(21,40)(22,32)(23,28)(24,30)(33,37)(34,39);(2,5)(6,8)(9,12,16,30,15,28,14,13)(10,18,32,39,20,37,21,19)(11,23,31,24,26,27,25,29)(17,33,40,34,35,36,22,38);(9,13,14,28,15,30,16,12)(10,19,21,37,20,39,32,18)(11,24,25,23,26,29,31,27)(17,34,22,33,35,38,40,36);(1,3,4,7)(2,6,5,8)(10,20)(17,35)(18,37)(19,39)(21,32)(22,40)(33,36)(34,38)

Polycyclic presentation:

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

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, 6363633);