#105915

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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

880 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,g^2=h,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, 13111);