#10143370

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	#49994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(34,38)(35,41)(40,43)(42,44);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,22)(20,28)(21,29)(23,27)(24,32)(25,30)(26,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,9)(3,4)(10,16)(11,12)(18,24)(19,27)(20,29)(26,30)(33,35,37,41)(34,39,38,46)(36,42,45,44)(40,47,43,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,23,29)(18,25,24,31)(19,28,27,21)(22,30,32,26)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,23,28)(18,26,24,30)(19,20,27,29)(22,25,32,31)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22,23,32)(18,27,24,19)(20,30,29,26)(21,31,28,25)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050566);