#415910

order	512
nilpotency class	3
isoclinism	isoclinic class of #394283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60415	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323106);