#239023

order	512
nilpotency class	3
isoclinism	isoclinic class of #239017
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,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, 146219);