#6671641

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

2099 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)(11,17)(12,24)(13,19)(14,30)(15,20)(16,21)(18,26)(22,35)(23,36)(25,33)(27,38)(28,40)(29,37)(31,34)(32,39);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,25)(13,27)(14,22)(15,31)(16,23)(18,37)(19,38)(20,34)(21,36)(24,33)(26,29)(28,32)(30,35)(39,40);(9,12,16,28)(10,18,21,38)(11,13,23,29)(14,27,15,26)(17,33,36,39)(19,34,37,35)(20,40,30,24)(22,32,31,25);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13,16,29)(10,19,21,37)(11,12,23,28)(14,25,15,32)(17,24,36,40)(18,34,38,35)(20,39,30,33)(22,26,31,27)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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, 6578837);