#500554

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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,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, 407750);