#10143354

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	#57336	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=i,e^2=gh,f^2=h,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, 10050550);