#219602

order	512
nilpotency class	3
isoclinism	isoclinic class of #219600
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

513 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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,d^b=di,f^b=fh,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, 126798);