#6487526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4057 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)(3,6)(4,7)(5,8)(9,10,15,18)(11,21,25,34)(12,26,28,33)(13,19,14,20)(16,23,31,24)(17,29,32,27)(22,30,38,40)(35,37,39,36);(3,4)(6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,31)(19,21)(20,34)(26,39)(27,36)(28,38)(29,37)(32,40)(33,35);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,32)(19,33)(20,26)(21,35)(23,36)(24,37)(25,38)(31,40)(34,39);(1,4,5,3)(2,7,8,6)(10,18)(16,31)(17,32)(19,20)(21,34)(26,33)(30,40)(35,39);(2,8)(6,7)(9,13,15,14)(10,19,18,20)(11,23,25,24)(12,27,28,29)(16,21,31,34)(17,33,32,26)(22,36,38,37)(30,35,40,39)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,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, 6394722);