#502410

order	512
nilpotency class	4
isoclinism	isoclinic class of #502410
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25339
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28246	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,d^2=e,e^2=i,f^2=hi,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=gi,f^c=fh,g^c=ghi,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, 409606);