#166183

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
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

1007 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,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,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73379);