#118525

order	512
nilpotency class	3
isoclinism	isoclinic class of #118521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ei,f^c=fh,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, 25721);