#240558

order	512
nilpotency class	3
isoclinism	isoclinic class of #10488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147754);