#107717

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

771 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)(3,7)(4,8)(9,10,13,19,15,18,14,20)(11,21,23,33,24,16,28,31)(12,25,27,32,26,34,29,17)(22,36,38,30,37,40,39,35);(1,2,5,3)(4,7,8,6)(9,11,14,23,15,24,13,28)(10,16,20,31,18,21,19,33)(12,22,27,39,26,37,29,38)(17,30,25,36,32,35,34,40);(1,3,5,2)(4,6,8,7)(9,12,15,26)(10,17,18,32)(11,22,24,37)(13,29,14,27)(16,30,21,35)(19,34,20,25)(23,39,28,38)(31,36,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14913);