#238939

order	512
nilpotency class	3
isoclinism	isoclinic class of #238935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

896 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)(3,7)(4,6)(5,9)(8,12)(10,13)(11,14)(15,16)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,30)(26,32);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(3,11)(7,14)(10,16)(13,15)

Polycyclic presentation:

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

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, 146135);