#8383490

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,22,21)(19,23,27,25)(20,28,30,24)(26,32,29,31)(33,34,39,36)(35,38,41,42)(37,40,47,44)(43,46,45,48);(2,7)(3,11)(4,14)(13,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,35)(34,38)(36,42)(37,45)(39,41)(40,48)(43,47)(44,46);(2,7)(5,9)(8,10)(13,16)(18,21)(23,25)(24,28)(31,32)(33,34,39,36)(35,42,41,38)(37,40,47,44)(43,48,45,46);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,36,39,34)(35,42,41,38)(37,40,47,44)(43,46,45,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,38,39,42)(34,41,36,35)(37,46,47,48)(40,45,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8290686);