#118867

order	512
nilpotency class	3
isoclinism	isoclinic class of #118862
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	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26063);