#116193

order	512
nilpotency class	3
isoclinism	isoclinic class of #116189
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

646 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fi,d^c=dh,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23389);