#233338

order	512
nilpotency class	3
isoclinism	isoclinic class of #233331
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	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1234 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,15)(13,16)(17,18)(19,29)(20,32)(21,26)(22,27)(23,28)(24,31)(25,30);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,24)(20,23)(21,30)(22,31)(25,28)(26,29)(27,32);(1,2)(3,12)(4,9)(5,8)(6,10)(7,14)(11,15)(13,16)(17,20,23,19)(18,25,28,24)(21,30,22,31)(26,29,27,32);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,11,16,12)(17,21,23,22)(18,26,28,27)(19,30,20,31)(24,29,25,32)

Polycyclic presentation:

2^9|c^2=i,d^2=h,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=fi,g^c=ghi

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, 140534);