#119177

order	512
nilpotency class	3
isoclinism	isoclinic class of #119168
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

678 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=d,d^2=i,e^2=i,f^2=h,g^2=hi,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, 26373);