#121858

order	512
nilpotency class	3
isoclinism	isoclinic class of #113366
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,7,4,6,10,5,9)(3,11,14,8,15,16,13,12)(17,18)(19,27,30,23)(20,31)(21,34)(22,25)(24,33)(26,36)(28,43,44,48)(29,38,47,42)(32,41)(35,40)(37,45,39,46);(1,3,7,13,6,15,5,14)(2,8,4,11,10,12,9,16)(17,19,21,29,22,30,36,47)(18,23,34,42,25,27,26,38)(20,28,35,46,32,44,33,45)(24,39,31,43,40,37,41,48);(1,4,7,10,6,9,5,2)(3,12,13,16,15,8,14,11)(17,20,21,35,22,32,36,33)(18,24,34,31,25,40,26,41)(19,28,29,46,30,44,47,45)(23,37,42,48,27,39,38,43)

Polycyclic presentation:

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

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, 29054);