#166023

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

775 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)(10,11)(12,13)(15,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,2)(3,6)(4,8)(5,7)(9,10,14,16)(11,12,15,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,16,11)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=g,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, 73219);