#240129

order	512
nilpotency class	3
isoclinism	isoclinic class of #240119
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 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)(3,4)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,18,22,26)(19,23,28,32)(20,25,21,24)(27,30,29,31);(1,3,6,11)(2,7,9,12)(4,8,13,14)(5,15,16,10)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,4,5,12,6,13,16,7)(2,3,8,10,9,11,14,15)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,30,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=di,f^c=fhi,g^c=gh

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, 147325);