#502690

order	512
nilpotency class	4
isoclinism	isoclinic class of #501096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

943 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,21,28)(12,16,22,29)(13,24,25,17)(19,26,32,37)(20,27,34,40)(23,36,35,30)(31,38,33,39);(1,3,4,8)(2,5,6,7)(11,19)(12,22)(13,25)(15,26)(16,29)(17,24)(20,33)(21,32)(27,39)(28,37)(31,34)(38,40);(9,11)(10,15)(12,20)(13,19,25,32)(14,21)(16,27)(17,26,24,37)(18,28)(22,34)(23,31,35,33)(29,40)(30,38,36,39);(2,6)(5,7)(9,12,14,22)(10,16,18,29)(11,20,21,34)(13,23,25,35)(15,27,28,40)(17,30,24,36)(19,31,32,33)(26,38,37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409886);