#8739220

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

4042 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)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,29)(20,26)(21,22)(23,28)(24,31)(25,32)(27,30)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,24)(20,21)(22,25)(23,31)(26,27)(28,29)(30,32)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(2,3)(7,9)(8,10)(15,16)(17,20,23,32)(18,25,28,26)(19,21,31,30)(22,29,27,24)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,26)(19,30)(21,31)(22,24)(23,32)(25,28)(27,29)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,23,30)(18,27,28,22)(19,20,31,32)(24,26,29,25)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,27)(18,21,28,30)(19,25,31,26)(20,29,32,24)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8646416);