#9717748

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

4057 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9624944);