#9050383

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8957579);