#119355

order	512
nilpotency class	3
isoclinism	isoclinic class of #119351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,g^c=gh

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, 26551);