#6684978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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	#61083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(11,17)(12,24)(13,19)(14,30)(15,20)(16,21)(18,23)(22,32)(25,40)(26,33)(27,35)(28,36)(29,38)(31,37)(34,39);(1,2)(3,6)(4,5)(7,8)(9,11,16,23)(10,17,21,18)(12,15,26,14)(13,27,28,25)(19,35,36,40)(20,33,30,24)(22,31,39,29)(32,37,34,38);(2,5)(6,7)(9,12,14,23,16,26,15,11)(10,18,20,24,21,17,30,33)(13,25,29,39,28,27,31,22)(19,34,37,40,36,32,38,35);(1,3,4,8)(2,6,5,7)(13,28)(19,36)(22,39)(25,27)(29,31)(32,34)(35,40)(37,38);(9,13)(10,19)(11,22)(12,25)(14,29)(15,31)(16,28)(17,32)(18,34)(20,37)(21,36)(23,39)(24,40)(26,27)(30,38)(33,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,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, 6592174);