#116220

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 23416);