#6494255

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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, 6401451);