#6355545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
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	C₄²	C₂⁵
prime radical	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1766 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,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, 6262741);