#510327

order	512
nilpotency class	4
isoclinism	isoclinic class of #510327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31495
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25434	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

466 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:

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

Polycyclic presentation:

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

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, 417523);