#239264

order	512
nilpotency class	3
isoclinism	isoclinic class of #238015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
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

924 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,22,26)(19,27,29,31)(20,25,21,24)(23,28,32,30);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(19,28)(20,21)(23,31)(24,25)(27,32)(29,30);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(19,29)(23,32)(27,31)(28,30)

Polycyclic presentation:

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

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, 146460);