#9123781

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	#59019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=gh,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, 9030977);