#273588

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

787 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)(3,12)(4,9)(5,8)(6,15)(7,11)(10,13)(14,16)(17,18)(19,27,30,23)(20,32,34,24)(21,26)(22,25)(28,36,40,31)(29,39)(33,41)(35,45,48,42)(37,38)(43,47)(44,46);(1,3,7,13)(2,8,11,16)(4,5,6,14)(9,10,15,12)(17,19,22,30)(18,23,25,27)(20,33,34,29)(21,36,38,31)(24,41,32,39)(26,28,37,40)(35,44,48,47)(42,46,45,43);(1,4,7,6)(2,9,11,15)(3,5,13,14)(8,10,16,12)(17,18)(19,28)(20,24)(21,37)(22,25)(23,36)(26,38)(27,31)(29,43)(30,40)(32,34)(33,46)(35,45)(39,47)(41,44)(42,48);(1,5,7,14)(2,10,11,12)(3,4,13,6)(8,9,16,15)(17,20,22,34)(18,24,25,32)(19,29,30,33)(21,35,38,48)(23,39,27,41)(26,42,37,45)(28,43,40,46)(31,44,36,47)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180784);