#479745

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

752 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)(9,10,14,19)(11,13)(12,17,27,34)(15,20,21,23)(16,25)(18,32)(22,29)(24,30)(26,37)(28,35,31,36)(33,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,21,24,14,25,15,18)(10,16,20,30,19,13,23,32)(12,22,31,33,27,37,28,38)(17,26,35,40,34,29,36,39);(2,6)(5,8)(9,12,14,27)(10,17,19,34)(11,22,25,37)(13,29,16,26)(15,31,21,28)(18,33,24,38)(20,36,23,35)(30,39,32,40);(3,7)(5,8)(9,13)(10,18)(11,23)(12,26)(14,16)(15,32)(17,33)(19,24)(20,25)(21,30)(22,35)(27,29)(28,40)(31,39)(34,38)(36,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,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, 386941);