#8873643

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#58064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780839);