#6494254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473174
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

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, 6401450);