#9161709

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

4505 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=hi,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^d=egi,f^e=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, 9068905);