#118869

order	512
nilpotency class	3
isoclinism	isoclinic class of #118868
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ 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:

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

Polycyclic presentation:

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

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, 26065);