#9039173

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

4204 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=gh,f^2=h,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, 8946369);