#219571

order	512
nilpotency class	3
isoclinism	isoclinic class of #219566
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

518 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126767);