#114358

order	512
nilpotency class	3
isoclinism	isoclinic class of #114344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
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

862 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh

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, 21554);