#107657

order	512
nilpotency class	3
isoclinism	isoclinic class of #107630
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

778 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)(3,7)(4,8)(9,10,13,19,15,20,14,18)(11,21,23,16,25,31,24,32)(12,26,27,34,28,33,29,17)(22,30,36,35,38,40,37,39);(1,2,5,3)(4,7,8,6)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,22,27,36,28,38,29,37)(17,30,33,39,34,40,26,35);(1,3,5,2)(4,6,8,7)(9,12,15,28)(10,17,20,34)(11,22,25,38)(13,29,14,27)(16,30,32,40)(18,26,19,33)(21,35,31,39)(23,37,24,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14853);