#505144

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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=fh,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, 412340);