#273701

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62091	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180897);