#502678

order	512
nilpotency class	4
isoclinism	isoclinic class of #501120
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	#24416	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

929 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,19)(13,16)(14,18)(15,29)(17,25)(20,27)(22,37)(23,30)(24,32)(26,34)(28,35)(31,38)(33,40)(36,39);(1,3,4,8)(2,5,6,7)(10,16)(11,22)(15,27)(17,31)(18,19)(21,36)(23,38)(24,39)(25,40)(26,28)(30,33)(32,37);(9,11,14,25,13,24,12,23)(10,17,19,21,16,30,18,32)(15,22,26,38,27,39,28,40)(20,33,34,36,29,31,35,37);(2,6)(5,7)(9,12,13,14)(10,18,16,19)(11,23,24,25)(15,26,27,28)(17,32,30,21)(20,34,29,35)(22,38,39,40)(31,37,33,36)

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,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, 409874);