#510260

order	512
nilpotency class	4
isoclinism	isoclinic class of #510259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31450
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,c^2=e,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417456);