#121948

order	512
nilpotency class	3
isoclinism	isoclinic class of #121711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

464 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)(12,16,24,32)(13,18)(14,17)(15,21)(20,29,38,37)(22,35)(23,31,39,40)(25,33,27,36)(26,30)(28,34);(2,5)(3,7)(9,11)(10,15,17,19)(12,23)(13,26)(14,21)(16,31,32,40)(18,35,34,30)(20,25)(22,28)(24,39)(27,38)(29,36,37,33);(2,5)(3,7)(9,12)(10,16,17,32)(11,20)(13,27)(14,24)(15,29,19,37)(18,36,34,33)(21,38)(22,39)(23,26)(25,28)(30,40,35,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29144);