#449565

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	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2100 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)(9,10,12,17,14,19,15,20)(11,21,22,31,24,16,25,33)(13,27,30,18,29,36,26,35)(23,38,39,34,28,40,37,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,28)(14,24)(15,25)(17,33)(18,34)(19,21)(20,31)(23,29)(26,39)(27,38)(30,37)(32,35)(36,40);(9,12,14,15)(10,17,19,20)(11,22,24,25)(13,26,29,30)(16,31,21,33)(18,27,35,36)(23,37,28,39)(32,38,34,40);(3,8)(5,7)(9,13,14,29)(10,18,19,35)(11,23,24,28)(12,26,15,30)(16,32,21,34)(17,27,20,36)(22,37,25,39)(31,38,33,40)

Polycyclic presentation:

2^9|a^2=eh,c^2=i,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, 356761);