#273256

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#3522	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 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)(3,10)(4,6)(5,9)(7,12)(8,14)(11,16)(13,15)(17,18)(19,28,31,41)(20,24)(21,26)(22,27)(23,30,25,34)(29,45,47,44)(32,39)(33,40)(35,42)(36,43)(37,46,38,48);(2,6)(4,14)(7,15)(8,9)(10,16)(11,12)(17,19)(18,23)(20,29)(21,34)(22,31)(24,37)(25,27)(26,41)(28,43)(30,36)(32,48)(33,47)(35,46)(38,40)(39,44)(42,45);(1,2)(3,7)(4,8)(5,9)(6,14)(10,12)(11,15)(13,16)(17,20,22,33)(18,24,27,40)(19,29,31,47)(21,35,36,32)(23,37,25,38)(26,42,43,39)(28,44,41,45)(30,48,34,46);(1,3,5,13)(2,7,9,16)(4,12,14,11)(6,15,8,10)(17,19)(18,25)(20,29)(21,34)(22,31)(23,27)(24,38)(26,28)(30,36)(32,48)(33,47)(35,46)(37,40)(39,45)(41,43)(42,44)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=i,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,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180452);