#9938067

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

4530 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)(3,5)(4,6)(7,8)(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26)(27,36)(28,30)(29,34)(31,33)(32,39)(35,40)(37,38);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27,31,39)(26,32,33,36)(28,34,35,37)(29,30,38,40);(10,17)(11,12)(18,24)(19,20)(26,33)(28,35)(32,36)(34,37);(1,2)(3,5)(4,6)(7,8)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,28,31,35)(26,30,33,40)(27,37,39,34)(29,36,38,32);(1,2)(3,7)(4,6)(5,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29)(26,34)(27,30)(28,36)(31,38)(32,35)(33,37)(39,40);(1,3,4,8)(2,5,6,7)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,30)(26,35)(27,38)(28,33)(29,39)(31,40)(32,37)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845263);