#9007888

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,e^2=ghi,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, 8915084);