#118689

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

689 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)(11,21,25,16)(12,26)(13,20)(14,19)(15,18)(17,28)(22,38,39,34)(23,32,24,33)(27,37,40,31)(29,36)(30,35);(9,11,15,25)(10,16,18,21)(12,27,29,40)(13,24,14,23)(17,34,35,38)(19,33,20,32)(22,30,39,28)(26,31,36,37);(9,12)(10,17,18,35)(11,22)(13,28)(14,30)(15,29)(16,31,21,37)(19,26,20,36)(23,27)(24,40)(25,39)(32,34,33,38)

Polycyclic presentation:

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

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, 25885);