#6685518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,21)(14,19)(16,18)(17,27)(20,24)(22,30)(25,36)(26,32)(28,29)(31,34)(33,39)(35,37)(38,40);(2,5)(3,7)(9,11,15,10)(12,23,26,32)(13,21,16,18)(14,29,30,34)(17,27,20,24)(19,31,22,28)(25,33,38,35)(36,37,40,39);(9,12,13,27,15,26,16,24)(10,17,18,23,11,20,21,32)(14,25,28,39,30,38,31,37)(19,33,34,36,22,35,29,40);(2,5)(6,8)(9,13,15,16)(10,18,11,21)(12,24,26,27)(14,28,30,31)(17,32,20,23)(19,34,22,29)(25,37,38,39)(33,40,35,36);(1,3,4,7)(2,6,5,8)(9,14,15,30)(10,19,11,22)(12,25,26,38)(13,28,16,31)(17,33,20,35)(18,34,21,29)(23,36,32,40)(24,37,27,39)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^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,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, 6592714);