#118348

order	512
nilpotency class	3
isoclinism	isoclinic class of #118344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

698 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25544);