#121644

order	512
nilpotency class	3
isoclinism	isoclinic class of #111253
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=dei,d^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dhi,e^c=ehi,g^c=gi

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, 28840);