#9040471

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

4535 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947667);