#504265

order	512
nilpotency class	4
isoclinism	isoclinic class of #504265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26464
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

504 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fh,d^2=e,e^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=gh,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, 411461);