#470511

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

664 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)(11,21)(12,26)(13,19)(14,20)(15,18)(16,24)(17,27)(22,31)(23,37)(25,32)(28,34)(29,33)(30,36)(35,39)(38,40);(9,11)(10,16,20,32)(12,23)(13,24)(14,22)(15,25)(17,30,33,37)(18,21,19,31)(26,35,34,40)(27,39)(28,36)(29,38);(1,3,4,8)(2,5,6,7)(11,22)(12,27)(13,15)(17,26)(18,19)(21,31)(23,38)(28,29)(30,35)(33,34)(36,39)(37,40);(9,12,15,29,14,28,13,27)(10,17,18,26,20,33,19,34)(11,23,25,38,22,36,24,39)(16,30,21,35,32,37,31,40)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=eg,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,d^c=de,e^c=ei,h^c=hi,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, 377707);