#6811502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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

2007 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=fi,e^2=fhi,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

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, 6718698);