#219593

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
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)(17,18,23,26)(19,22,29,38)(20,30)(21,28,37,34)(24,33)(25,40,42,36)(27,32)(31,45)(35,43)(39,48)(41,47)(44,46);(17,19,22,26,23,29,38,18)(20,30,46,27,33,24,32,44)(21,34,36,25,37,28,40,42)(31,39,48,45,35,41,47,43);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,20,22,32,23,33,38,46)(18,24,29,44,26,30,19,27)(21,35,36,48,37,31,40,47)(25,41,34,43,42,39,28,45);(3,10)(6,15)(9,16)(11,14)(17,21,22,36,23,37,38,40)(18,25,19,28,26,42,29,34)(20,31,32,47,33,35,46,48)(24,39,27,43,30,41,44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,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, 126789);