#503985

order	512
nilpotency class	4
isoclinism	isoclinic class of #503983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26232
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28504	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=ei,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,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,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, 411181);