#472563

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

790 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)(11,20)(12,25)(13,18)(14,29)(15,19)(16,23)(17,27)(21,32)(22,37)(24,30)(26,33)(28,34)(31,38)(35,36)(39,40);(9,11)(10,16,19,20)(12,22)(13,21)(14,24)(15,23)(17,31,33,35)(18,30,29,32)(25,40,34,37)(26,38)(27,36)(28,39);(1,3,4,8)(2,5,6,7)(11,21,23,24)(16,30,20,32)(22,38,39,36)(31,40,35,37);(9,12,15,28)(10,17,19,33)(11,22,23,39)(13,27,14,26)(16,31,20,35)(18,34,29,25)(21,36,24,38)(30,37,32,40)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^d=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379759);