#9159953

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9067149);