#119356

order	512
nilpotency class	3
isoclinism	isoclinic class of #119351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26552);