#501531

order	512
nilpotency class	4
isoclinism	isoclinic class of #501530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24658
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=ehi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi,h^c=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, 408727);