#233893

order	512
nilpotency class	3
isoclinism	isoclinic class of #233889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
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

1312 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

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, 141089);