#9039375

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

4520 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(27,28)(30,34)(35,36)(37,39);(1,2,6,5)(3,8,7,4)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,23)(22,24)(25,26,31,33)(27,34,28,30)(29,32,38,40)(35,39,36,37);(1,3)(2,4)(5,8)(6,7)(10,15)(12,16)(13,22)(19,24)(25,27,31,28)(26,30,33,34)(29,36,38,35)(32,39,40,37);(1,2,6,5)(3,4,7,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28,31,27)(26,30,33,34)(29,35,38,36)(32,39,40,37);(1,4,6,8)(2,7,5,3)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,38)(26,32,33,40)(27,35,28,36)(30,37,34,39);(1,5,6,2)(3,8,7,4)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,27)(28,33)(29,37)(31,34)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946571);