#445122

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 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)(9,10,15,20,16,21,14,19)(11,22,12,17,25,32,24,33)(13,27,28,18,29,35,30,36)(23,38,26,37,39,31,40,34);(9,11,16,25)(10,17,21,33)(12,15,24,14)(13,26,29,40)(18,34,36,37)(19,32,20,22)(23,30,39,28)(27,38,35,31);(2,5)(6,8)(9,12,15,25,16,24,14,11)(10,17,20,32,21,33,19,22)(13,26,28,39,29,40,30,23)(18,31,35,34,36,38,27,37);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,23)(12,26)(14,30)(15,28)(16,29)(17,31)(19,27)(20,35)(21,36)(22,37)(24,40)(25,39)(32,34)(33,38)

Polycyclic presentation:

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

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, 352318);