#119169

order	512
nilpotency class	3
isoclinism	isoclinic class of #119168
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

592 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26365);