#9159951

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	#51938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 9067147);