#449563

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

2044 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)(3,7,8,5)(9,10,15,29,13,18,14,19)(11,20,30,32,22,16,23,36)(12,24,28,17,26,35,27,34)(21,37,39,33,25,40,38,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,22)(14,23)(15,30)(17,33)(18,20)(19,32)(21,26)(24,37)(27,39)(28,38)(29,36)(31,34)(35,40);(16,20)(31,33)(32,36)(37,40);(3,8)(5,7)(9,12,13,26)(10,17,18,34)(11,21,22,25)(14,28,15,27)(16,31,20,33)(19,35,29,24)(23,39,30,38)(32,37,36,40)

Polycyclic presentation:

2^9|a^2=eh,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,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, 356759);