#6684992

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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
prime radical	#61082	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,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,d^c=di,h^c=hi,e^d=ei,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, 6592188);