#500550

order	512
nilpotency class	4
isoclinism	isoclinic class of #499855
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	C₂×#454	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

874 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)(4,5)(6,8)(11,21)(13,26)(16,31)(18,27)(22,37)(23,38)(24,28)(25,30)(29,39)(32,34)(33,36)(35,40);(2,5)(6,7)(9,10,15,20)(11,22,24,35)(12,19,14,17)(13,27,26,18)(16,29,32,23)(21,37,28,40)(25,33,30,36)(31,39,34,38);(9,11)(10,16)(12,23)(13,28)(14,29)(15,24)(17,22)(18,34)(19,35)(20,32)(21,26)(25,39)(27,31)(30,38)(33,40)(36,37);(1,3,4,8)(2,6,5,7)(9,12,15,14)(10,17,20,19)(11,23,24,29)(13,25,26,30)(16,22,32,35)(18,33,27,36)(21,38,28,39)(31,37,34,40)

Polycyclic presentation:

2^9|b^2=i,d^2=ei,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,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407746);