#449564

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

Polycyclic presentation:

2^9|a^2=eh,c^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.

Relationships

maximal subgroup of	#72286066, #72286082, #72286224
maximal normal subgroup of	#72286066, #72286082, #72286224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356760);