#8873640

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

3939 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,16)(13,15)(17,18,23,22)(19,24,28,27)(20,29,31,25)(21,32,30,26)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,5,11)(2,6,9,7)(4,13,14,10)(8,16,12,15)(18,22)(24,27)(25,29)(26,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,16)(12,15)(13,14)(17,19,23,28)(18,24,22,27)(20,30,31,21)(25,32,29,26)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,20,23,31)(18,25,22,29)(19,21,28,30)(24,26,27,32)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,16)(12,15)(13,14)(17,21)(18,26)(19,20)(22,32)(23,30)(24,25)(27,29)(28,31)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,4,5,14)(2,8,9,12)(3,10,11,13)(6,15,7,16)(17,22,23,18)(19,27,28,24)(20,29,31,25)(21,32,30,26)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=ghi,e^2=hi,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, 8780836);