#105931

order	512
nilpotency class	3
isoclinism	isoclinic class of #105909
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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

857 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)(11,21,23,16,24,32,30,33)(12,25,28,17,29,35,27,36)(22,34,38,40,26,31,39,37);(1,2,5,7,4,8,3,6)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,32)(19,21)(20,33)(22,29)(25,37)(27,38)(28,39)(31,36)(35,40);(1,3,4,5)(2,7,8,6)(9,12,15,29)(10,17,20,36)(11,22,24,26)(13,28,14,27)(16,31,33,34)(18,25,19,35)(21,37,32,40)(23,39,30,38)

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,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13127);