#9064350

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
prime radical	#43901	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4507 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=gh,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, 8971546);