#9065909

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

4618 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973105);