#451840

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

787 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)(3,5,7,8)(9,10,14,18)(11,20)(12,25,27,32)(13,16,28,19)(15,24)(17,33,35,26)(21,36)(22,31)(23,29)(30,37)(34,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,22)(14,24)(16,29)(17,34)(18,20)(19,31)(23,28)(25,40)(26,37)(27,39)(30,35)(32,36)(33,38);(3,7)(5,8)(10,16,18,19)(15,29,20,31)(17,32,35,25)(30,40,34,36);(9,12,13,26,14,27,28,33)(10,17,19,25,18,35,16,32)(11,21,22,37,24,39,23,38)(15,30,31,36,20,34,29,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359036);