#8720654

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
prime radical	#46687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627850);