#501873

order	512
nilpotency class	4
isoclinism	isoclinic class of #501872
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24913
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33533	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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)(3,14)(4,8)(5,18)(6,16)(7,12)(9,17)(10,23)(11,27)(13,26)(15,21)(19,25)(20,31)(22,32)(24,29)(28,30);(1,3,7,29,8,18,26,19)(2,9,13,15,4,20,12,23)(5,22,24,30,14,11,25,16)(6,10,27,31,28,21,32,17);(1,4)(2,8)(3,15)(5,21)(6,11)(7,12)(9,19)(10,14)(13,26)(16,27)(17,24)(18,23)(20,29)(22,28)(25,31)(30,32);(1,5,8,14)(2,10,4,21)(3,16,18,30)(6,20,28,9)(7,24,26,25)(11,29,22,19)(12,17,13,31)(15,27,23,32)

Polycyclic presentation:

2^9|b^2=fh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409069);