#6467843

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(4,5)(7,8)(9,10)(11,20)(12,17)(13,28)(14,18)(15,19)(16,22)(21,36)(23,31)(24,30)(25,39)(26,34)(27,35)(29,32)(33,37)(38,40);(9,11,13,22,15,23,14,29)(10,16,18,20,19,32,28,31)(12,24,25,37,27,38,26,21)(17,33,34,30,35,36,39,40);(2,5)(6,8)(9,12,13,25,15,27,14,26)(10,17,28,39,19,35,18,34)(11,21,29,38,23,37,22,24)(16,30,20,36,32,40,31,33);(9,12,13,25,15,27,14,26)(10,17,28,39,19,35,18,34)(11,21,29,38,23,37,22,24)(16,30,20,36,32,40,31,33);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 6375039);