#243048

order	512
nilpotency class	3
isoclinism	isoclinic class of #243042
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 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:

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

Polycyclic presentation:

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

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, 150244);