#243006

order	512
nilpotency class	3
isoclinism	isoclinic class of #243002
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150202);