#482080

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60458	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1048 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)(3,7,8,5)(4,6)(9,10)(11,20)(12,15)(13,27)(14,19)(16,21)(17,24)(18,25)(22,33)(23,32)(26,30)(28,36)(29,39)(31,40)(34,38)(35,37);(1,3)(2,5)(4,8)(6,7)(10,15)(12,24)(13,25)(16,28)(17,18)(19,30)(20,33)(22,38)(23,37)(26,39)(27,29)(31,32)(34,35)(36,40);(2,6)(5,7)(9,11)(10,16)(12,22)(13,23)(14,21)(15,28)(17,31)(18,32)(19,20)(24,38)(25,37)(26,36)(27,35)(29,34)(30,33)(39,40);(11,21)(16,20)(22,36)(23,37)(28,33)(31,34)(32,35)(38,40)

Polycyclic presentation:

2^9|a^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

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, 389276);