#105929

order	512
nilpotency class	3
isoclinism	isoclinic class of #105908
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fi

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, 13125);