#9138440

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

4585 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)(11,16)(12,14)(13,20)(15,19)(17,22)(18,23)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,39,38,40);(1,2)(3,4)(5,7)(6,8)(9,11,14,19)(10,15,12,16)(13,18,22,21)(17,23,20,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,40)(37,38);(1,3,5,8)(2,6,7,4)(10,12)(15,16)(17,20)(23,24)(28,37)(32,36)(34,38)(39,40);(3,8)(4,6)(9,12,14,10)(11,16,19,15)(13,20,22,17)(18,24,21,23)(25,28,30,37)(26,32,29,36)(27,34,35,38)(31,39,33,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,27)(26,33)(28,38)(29,31)(30,35)(32,39)(34,37)(36,40);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,40,38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^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, 9045636);