#6671347

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25193	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578543);