#6806824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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	#60764	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1929 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,28)(13,19)(14,18)(15,21)(16,25)(17,24)(20,29)(23,37)(26,36)(27,40)(30,39)(31,35)(32,34)(33,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,25)(10,16,18,22)(12,13,30,32)(15,26,29,27)(17,37,35,38)(19,28,34,39)(20,36,21,40)(23,24,33,31);(9,12,29,24,14,30,15,31)(10,17,20,39,18,35,21,28)(11,23,26,32,25,33,27,13)(16,34,40,38,22,19,36,37);(2,6)(5,7)(10,18)(16,22)(17,35)(19,34)(20,21)(28,39)(36,40)(37,38);(3,8)(5,7)(9,13,29,23,14,32,15,33)(10,19,20,37,18,34,21,38)(11,24,26,30,25,31,27,12)(16,35,40,28,22,17,36,39)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=hi,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, 6714020);