#506198

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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)(4,6)(5,8)(9,10)(11,15)(12,17)(13,19)(14,20)(16,22)(18,26)(21,29)(23,30)(24,31)(25,32)(27,33)(28,36)(34,37)(35,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11,13,23,26,21,37,38,14,24,28,39,12,22,25,27)(10,15,19,30,18,29,34,40,20,31,36,35,17,16,32,33);(9,11)(10,16)(12,21)(13,27)(14,24)(15,17)(18,31)(19,35)(20,29)(22,26)(23,25)(28,38)(30,36)(32,33)(34,40)(37,39);(2,6)(5,7)(9,12,14,26)(10,17,20,18)(11,21,24,22)(13,25,28,37)(15,29,31,16)(19,32,36,34)(23,38,39,27)(30,40,35,33)

Polycyclic presentation:

2^9|b^2=eg,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413394);