#273498

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61220	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

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, 180694);