#118871

order	512
nilpotency class	3
isoclinism	isoclinic class of #118868
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,6,8,5)(4,7)(9,10,14,20,15,19,13,18)(11,21,24,16,25,33,23,32)(12,26,28,36,29,17,30,35)(22,38,39,31,27,37,40,34);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,19,32)(12,27,29,22)(13,24,14,23)(17,34,26,31)(18,33,20,21)(28,39,30,40)(35,37,36,38);(1,3)(2,6,7,5)(4,8)(9,12,13,28,15,29,14,30)(10,17,18,35,19,26,20,36)(11,22,23,39,25,27,24,40)(16,31,21,37,32,34,33,38)

Polycyclic presentation:

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

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, 26067);