#499569

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=e,d^2=i,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406765);