#116718

order	512
nilpotency class	3
isoclinism	isoclinic class of #116714
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

621 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,f^b=fi,e^c=eh,f^c=fi,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, 23914);