#118671

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25867);