#9016447

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

4390 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8923643);