#433079

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 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)(3,5)(4,6)(7,8)(9,10,15,20,16,22,14,21)(11,23,25,17,27,36,26,37)(12,28,30,18,32,39,31,35)(13,19,24,34,29,38,33,40);(1,3,4,7)(2,5,6,8)(9,11,14,26,16,27,15,25)(10,17,21,23,22,37,20,36)(12,29,31,24,32,13,30,33)(18,38,28,34,35,19,39,40);(2,6)(5,8)(9,12)(10,18)(11,24)(13,25)(14,30)(15,31)(16,32)(17,34)(19,36)(20,28)(21,39)(22,35)(23,38)(26,29)(27,33)(37,40);(9,13,14,33,16,29,15,24)(10,19,20,34,22,38,21,40)(11,12,25,30,27,32,26,31)(17,35,23,39,37,18,36,28)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,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, 340275);