#479739

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386935);