#6459453

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#60359	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366649);