#510116

order	512
nilpotency class	4
isoclinism	isoclinic class of #510115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31314
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25805	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,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=ghi,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, 417312);