#219564

order	512
nilpotency class	3
isoclinism	isoclinic class of #219552
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

528 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,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,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126760);