#233868

order	512
nilpotency class	3
isoclinism	isoclinic class of #233856
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=i,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,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,e^c=ei,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, 141064);