#316148

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=eh,c^2=fhi,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh

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, 223344);