#441106

order	512
nilpotency class	3
isoclinism	isoclinic class of #420407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348302);