#501930

order	512
nilpotency class	4
isoclinism	isoclinic class of #501930
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24965
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409126);