#9138456

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

4412 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)(27,28)(29,38)(30,33)(31,39)(32,36)(34,37)(35,40);(1,2)(3,4)(5,7)(6,8)(9,11,14,18)(10,15,19,13)(12,21,23,20)(16,22,17,24)(25,27,32,33)(26,30,36,28)(29,31,37,40)(34,39,38,35);(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)(26,33,36,27)(29,39,37,35)(31,38,40,34);(3,8)(4,6)(9,12)(10,17)(11,21)(13,22)(14,23)(15,24)(16,19)(18,20)(25,29,32,37)(26,34,36,38)(27,31,33,40)(28,35,30,39);(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)(26,33,36,27)(29,35,37,39)(31,38,40,34);(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)(26,35)(27,37)(28,38)(29,33)(30,34)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=gh,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, 9045652);