#479741

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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)(9,10,16,19,14,20,15,30)(11,21,25,34,24,17,31,36)(12,18,29,35,13,27,26,37)(22,32,23,39,40,38,28,33);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17,20,21)(12,23)(13,28)(14,24)(15,31)(16,25)(18,33,27,39)(19,36,30,34)(22,26)(29,40)(32,37,38,35);(2,6)(5,8)(9,12,14,13)(10,18,20,27)(11,22,24,40)(15,29,16,26)(17,32,21,38)(19,37,30,35)(23,25,28,31)(33,34,39,36);(3,7)(5,8)(9,13,14,12)(10,18,20,27)(11,23,24,28)(15,26,16,29)(17,33,21,39)(19,37,30,35)(22,31,40,25)(32,36,38,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=i,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, 386937);