#121761

order	512
nilpotency class	3
isoclinism	isoclinic class of #112870
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10)(11,19,22,32)(12,16)(13,17)(14,18)(15,21,28,25)(20,35,37,34)(23,30)(24,31)(26,33)(27,38,29,36)(39,40);(2,5)(3,7)(9,11,13,25,14,22,26,21)(10,15,17,32,18,28,33,19)(12,20,23,38,24,37,39,36)(16,29,30,34,31,27,40,35);(2,5)(3,7)(9,12,13,23,14,24,26,39)(10,16,17,30,18,31,33,40)(11,20,25,38,22,37,21,36)(15,27,32,35,28,29,19,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28957);