#502879

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
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

849 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)(4,5)(6,8)(9,10,16,21)(11,17,25,35)(12,26,18,28)(13,29,32,30)(14,20,15,19)(22,23,39,31)(24,37,33,34)(27,38,36,40);(2,5)(6,7)(9,11)(10,17)(12,27)(13,31)(14,33)(15,24)(16,25)(18,36)(19,37)(20,34)(21,35)(22,29)(23,32)(26,38)(28,40)(30,39);(9,12,28,10,15,30,13,20,16,18,26,21,14,29,32,19)(11,22,38,34,24,27,23,35,25,39,40,37,33,36,31,17);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410075);