#121495

order	512
nilpotency class	3
isoclinism	isoclinic class of #9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

495 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)(3,9)(4,13)(5,8)(6,11)(7,14)(10,16)(12,15)(17,18,21,26,23,27,22,25)(19,28,29,20,24,31,30,32);(1,3,4,11,5,12,14,16)(2,6,7,9,8,10,13,15)(17,19,21,29,23,24,22,30)(18,20,25,28,27,32,26,31);(1,2)(3,10,12,6)(4,7)(5,8)(9,11,15,16)(13,14)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,29)(26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28691);