#119673

order	512
nilpotency class	3
isoclinism	isoclinic class of #119665
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fh,e^c=ei,g^c=gh

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, 26869);