#8183773

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

4147 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(34,38)(39,41)(40,45)(46,47);(2,9)(6,10)(8,11)(13,16)(18,22)(23,29)(24,27)(30,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

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

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, 8090969);