#9065801

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

4425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^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, 8972997);