#233336

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1250 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,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, 140532);