#503371

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
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×Q₈	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 410567);