#6491469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2921 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)(9,10,16,19,15,21,14,20)(11,22,25,17,27,36,26,32)(12,28,29,34,13,18,30,35)(23,37,40,33,24,38,39,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,23)(14,25)(15,27)(16,26)(18,33)(19,22)(20,36)(21,32)(28,31)(29,40)(30,39)(34,38)(35,37);(2,5)(6,8)(9,12,15,13)(10,18,21,28)(11,23,27,24)(14,29,16,30)(17,31,32,33)(19,34,20,35)(22,37,36,38)(25,39,26,40);(2,5)(6,8)(9,13,15,12)(10,18,21,28)(11,24,27,23)(14,29,16,30)(17,31,32,33)(19,35,20,34)(22,38,36,37)(25,39,26,40);(1,3,4,7)(2,6,5,8)(9,14,15,16)(10,19,21,20)(11,25,27,26)(12,29,13,30)(17,22,32,36)(18,34,28,35)(23,39,24,40)(31,37,33,38)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,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, 6398665);