#502656

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

781 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)(9,10,16,21)(11,17,26,22)(12,14,18,28)(13,19,31,29)(15,33,30,20)(23,27,32,37)(24,39,35,36)(25,40,34,38);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,27)(13,24)(14,32)(15,34)(16,26)(18,37)(19,39)(20,40)(21,22)(23,28)(25,30)(29,36)(31,35)(33,38);(9,12,28,10,15,29,13,20,16,18,14,21,30,19,31,33)(11,22,24,27,25,40,23,39,26,17,35,37,34,38,32,36);(2,6)(5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409852);