#6447133

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2061 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)(4,8)(5,7)(9,10,14,20,15,19,28,37)(11,21,24,35,25,40,32,16)(12,26,13,18,29,17,31,38)(22,39,30,33,27,34,23,36);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,27)(13,30)(14,32)(15,25)(17,36)(18,39)(19,35)(20,40)(21,37)(22,29)(23,31)(24,28)(26,33)(34,38);(1,4,5,3)(2,6,7,8)(9,12,28,31,15,29,14,13)(10,17,37,18,19,26,20,38)(11,22,32,23,25,27,24,30)(16,33,40,39,35,36,21,34);(9,13,14,29,15,31,28,12)(10,18,20,17,19,38,37,26)(11,23,24,22,25,30,32,27)(16,34,21,36,35,39,40,33);(10,19)(16,35)(17,26)(18,38)(20,37)(21,40)(33,36)(34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fg,d^2=f,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, 6354329);