#6671639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2104 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,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,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578835);