#6438158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345354);