#6343044

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=di,f^a=fh,c^b=ci,f^b=fh,e^c=eh,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250240);