#458517

order	512
nilpotency class	3
isoclinism	isoclinic class of #458517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16319
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365713);