#9123893

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

4454 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)(11,15)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26,32,36)(27,30,33,28)(29,39,38,34)(31,40,37,35);(1,2)(3,6)(4,7)(5,8)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,27)(26,28)(29,37)(30,36)(31,38)(32,33)(34,35)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10,14,18)(11,19,20,15)(12,16,23,22)(13,17,21,24)(25,28,32,30)(26,33,36,27)(29,40,38,35)(31,34,37,39);(3,8)(5,6)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,29,32,38)(26,34,36,39)(27,37,33,31)(28,35,30,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,23)(10,16,18,22)(11,13,20,21)(15,24,19,17)(25,30,32,28)(26,33,36,27)(29,35,38,40)(31,34,37,39);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)(25,31,32,37)(26,35,36,40)(27,38,33,29)(28,34,30,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=g,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, 9031089);