#106429

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

892 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci,d^b=dh,f^b=fh,e^c=eh,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, 13625);