#273785

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
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	#62238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180981);