#505374

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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)(11,16)(12,18)(13,19)(14,17)(15,20)(21,30)(22,31)(23,32)(24,40)(25,34)(26,35)(27,38)(28,36)(29,37)(33,39);(2,5)(6,7)(9,11,21,25,13,26,15,22)(10,16,30,34,19,35,20,31)(12,23,29,24,14,28,27,39)(17,36,38,33,18,32,37,40);(9,12)(10,17)(11,22)(13,14)(15,29)(16,31)(18,19)(20,38)(21,27)(24,39)(25,26)(30,37)(33,40)(34,35);(1,3,4,8)(2,6,5,7)(9,12,21,29,13,14,15,27)(10,18,30,37,19,17,20,38)(11,23,25,24,26,28,22,39)(16,32,34,40,35,36,31,33)

Polycyclic presentation:

2^9|b^2=fh,d^2=efh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,h^c=hi,g^d=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, 412570);