#8934326

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

4476 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841522);