#121775

order	512
nilpotency class	3
isoclinism	isoclinic class of #112946
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

421 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,f^c=fh,g^c=gh

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, 28971);