#237441

order	512
nilpotency class	3
isoclinism	isoclinic class of #237435
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	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1440 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)(3,10)(4,5)(9,12)(11,13)(14,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,2,6,9)(3,8,11,15)(4,12,5,7)(10,16,13,14)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,3,6,11)(2,8,9,15)(4,10,5,13)(7,14,12,16)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30);(3,11)(8,15)(10,13)(14,16)(19,28)(23,31)(26,29)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144637);