#119309

order	512
nilpotency class	3
isoclinism	isoclinic class of #119300
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

546 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26505);