#168415

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1001 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75611);