#9138455

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	#43043	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4454 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=g,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, 9045651);