#239027

order	512
nilpotency class	3
isoclinism	isoclinic class of #239025
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	#60916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,d^c=di,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, 146223);