#9017237

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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:

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

Polycyclic presentation:

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

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, 8924433);