#109413

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,6,10,5,9)(3,11,12,15,13,16,14,7)(17,18)(19,28,31,24)(20,33)(21,37)(22,27)(23,26)(25,34)(29,43,45,47)(30,39,32,42)(35,40)(36,48)(38,44,41,46);(1,3,5,12,6,13,4,14)(2,7,9,11,10,15,8,16)(17,19,22,32,23,31,21,30)(18,24,27,39,26,28,37,42)(20,29,35,46,36,45,34,44)(25,38,33,47,40,41,48,43);(17,20,22,35,23,36,21,34)(18,25,37,48,26,40,27,33)(19,29,32,46,31,45,30,44)(24,38,42,43,28,41,39,47)

Polycyclic presentation:

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

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, 16609);