#6671003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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	#3012	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(9,10,15,20)(11,16,24,33)(12,25,28,18)(13,17,14,19)(21,31,23,32)(22,37,39,30)(26,40,38,35)(27,36,29,34);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,23)(14,21)(15,24)(17,32)(18,35)(19,31)(20,33)(22,27)(25,40)(28,38)(29,39)(30,34)(36,37);(2,5)(6,7)(10,17)(11,21)(13,14)(18,34)(19,20)(22,26)(23,24)(25,36)(27,29)(31,32)(35,40)(38,39);(1,3,4,8)(2,6,5,7);(9,12,15,28)(10,18,20,25)(11,22,24,39)(13,27,14,29)(16,30,33,37)(17,34,19,36)(21,26,23,38)(31,35,32,40)

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,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, 6578199);