#236278

order	512
nilpotency class	3
isoclinism	isoclinic class of #236265
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1243 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)(3,7,10,16)(4,6,12,8)(11,15,13,14)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(2,6)(4,12)(7,15)(8,9)(11,13)(14,16)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(1,3)(2,7)(4,13)(5,10)(6,14)(8,15)(9,16)(11,12)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,27)(29,31);(3,10)(7,16)(11,13)(14,15)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

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, 143474);