#469333

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,23)(13,26)(14,18)(15,21)(16,25)(17,27)(20,34)(22,32)(24,31)(28,29)(30,40)(33,36)(35,37)(38,39);(2,5)(6,8)(13,27)(15,29)(17,26)(21,28)(24,39)(30,37)(31,38)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,22)(10,15,18,29)(12,20,25,36)(13,28,27,21)(16,30,23,37)(17,32,26,19)(24,40,39,35)(31,34,38,33);(3,7)(6,8)(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,30,29,37)(17,31,26,38)(19,33,32,34)(21,35)(22,36)(27,39)(28,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376529);