#6671655

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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

2058 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)(9,10,15,21)(11,16,26,36)(12,27,23,37)(13,30,22,18)(14,20,33,19)(17,31,38,29)(24,40,25,35)(28,39,32,34);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,28)(13,31)(14,24)(15,26)(17,18)(19,35)(20,40)(21,36)(22,29)(23,32)(25,33)(27,39)(30,38)(34,37);(9,12,15,23)(10,17,21,38)(11,22,26,13)(14,31,33,29)(16,34,36,39)(18,35,30,40)(19,37,20,27)(24,28,25,32);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13,15,22)(10,18,21,30)(11,23,26,12)(14,28,33,32)(16,27,36,37)(17,35,38,40)(19,39,20,34)(24,31,25,29)

Polycyclic presentation:

2^9|a^2=i,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,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, 6578851);