#479747

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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)(11,21,26,27)(12,15)(13,29)(14,19)(16,20)(17,24,22,25)(18,31)(23,35,39,34)(28,33)(30,32)(36,38,40,37);(1,3)(2,5)(4,7)(6,8)(9,11,14,24,16,26,15,25)(10,17,12,27,20,22,19,21)(13,23,31,37,33,39,32,38)(18,35,29,40,30,34,28,36);(3,7)(5,8)(9,12,14,10,16,19,15,20)(11,22,24,27,26,17,25,21)(13,30,31,29,33,18,32,28)(23,36,37,34,39,40,38,35);(9,13,16,33)(10,18,20,30)(11,23,26,39)(12,28,19,29)(14,32,15,31)(17,34,22,35)(21,36,27,40)(24,38,25,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^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,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

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, 386943);