#9938140

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

4348 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,16)(14,22)(15,17)(18,23)(19,24)(20,21)(25,26,31,33)(27,32,36,29)(28,39,38,34)(30,40,37,35);(1,3)(2,6)(4,5)(7,8)(9,11,15,13)(10,16,17,12)(14,19,23,21)(18,24,22,20)(25,27,31,36)(26,32,33,29)(28,30,38,37)(34,35,39,40);(10,17)(12,16)(18,22)(20,24)(26,33)(27,36)(28,38)(35,40);(1,2)(3,8)(4,6)(5,7)(9,12,15,16)(10,11,17,13)(14,20,23,24)(18,19,22,21)(25,28,31,38)(26,34,33,39)(27,37,36,30)(29,35,32,40);(1,4,5,3)(2,6,7,8)(9,13,15,11)(10,16,17,12)(14,19,23,21)(18,20,22,24)(25,29)(26,27)(28,35)(30,39)(31,32)(33,36)(34,37)(38,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,23)(10,18,17,22)(11,19,13,21)(12,20,16,24)(25,30,31,37)(26,35,33,40)(27,38,36,28)(29,34,32,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=g,f^2=h,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, 9845336);