#168658

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75854);