#475828

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383024);