#203138

order	512
nilpotency class	3
isoclinism	isoclinic class of #19368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110334);