#9717782

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

4527 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)(3,4)(5,6)(7,8)(9,10)(11,13)(12,19)(14,22)(15,16)(17,24)(18,23)(20,21)(28,36)(31,38)(34,40)(37,39);(2,6)(3,7)(9,11,15,12)(10,13,16,19)(14,18,24,20)(17,21,22,23)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(1,3,5,7)(2,4,6,8)(10,16)(11,12)(17,22)(18,20)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4)(2,7)(3,6)(5,8)(9,12,15,11)(10,13,16,19)(14,20,24,18)(17,21,22,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(9,13)(10,11)(12,16)(14,23)(15,19)(17,20)(18,22)(21,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(9,14,15,24)(10,17,16,22)(11,18,12,20)(13,21,19,23)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624978);