#8180394

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

4142 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=h,f^2=gh,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, 8087590);