#9015534

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

4626 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=ghi,e^2=i,f^2=h,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, 8922730);