#466886

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

725 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 374082);