#512216

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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)(3,6)(4,5)(7,8)(10,17)(11,20)(12,25)(13,18)(14,30)(15,22)(19,23)(21,33)(24,32)(27,34)(28,29)(31,35)(36,38)(37,39);(2,5)(6,8)(9,10,18,24,15,20,30,33,16,21,14,11,22,32,13,17)(12,19,34,39,28,36,40,38,29,37,27,23,25,35,26,31);(1,3,4,7)(2,6,5,8)(9,11,13,21,22,33,14,20,16,24,30,10,15,17,18,32)(12,23,26,37,25,38,27,36,29,39,40,19,28,31,34,35);(9,12)(10,19)(11,23)(13,26)(14,27)(15,28)(16,29)(17,31)(18,34)(20,36)(21,37)(22,25)(24,39)(30,40)(32,35)(33,38)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=ci,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419412);