#501659

order	512
nilpotency class	4
isoclinism	isoclinic class of #501658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24750
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

653 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fh,g^c=gi,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, 408855);