#119945

order	512
nilpotency class	3
isoclinism	isoclinic class of #119940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

529 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=i,f^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=dh,e^b=ei,f^b=fi,e^c=ehi,f^c=fhi,g^c=ghi

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, 27141);