#119383

order	512
nilpotency class	3
isoclinism	isoclinic class of #119380
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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

855 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)(11,21,25,16,24,32,23,33)(12,26,28,36,29,17,30,35)(22,37,27,40,38,34,39,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,25)(14,23)(15,24)(17,34)(18,32)(19,33)(20,21)(22,28)(26,37)(29,39)(30,38)(31,36)(35,40);(1,3)(2,6,7,5)(4,8)(9,12,13,28,15,29,14,30)(10,17,18,35,19,26,20,36)(11,22,23,27,24,38,25,39)(16,31,21,37,33,40,32,34)

Polycyclic presentation:

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

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, 26579);