#6671647

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

2089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,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,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, 6578843);