#9713371

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

4216 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)(3,4)(5,8)(6,7)(9,10)(11,17)(12,13)(14,16)(15,21)(18,23)(19,22)(20,24)(28,36)(31,38)(35,39)(37,40);(1,3)(2,4)(5,6)(7,8)(9,11,14,21)(10,15,16,17)(12,19,18,24)(13,20,23,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,2,7,6)(3,5,8,4)(10,16)(12,18)(15,17)(19,24)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,37,38,40);(1,4,7,5)(2,8,6,3)(9,12,14,18)(10,13,16,23)(11,19,21,24)(15,20,17,22)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,38)(34,40);(1,5,7,4)(2,8,6,3)(9,11)(10,17)(12,19)(13,22)(14,21)(15,16)(18,24)(20,23)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(1,6,7,2)(3,5,8,4)(9,13)(10,18)(11,20)(12,16)(14,23)(15,24)(17,19)(21,22)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,38)(36,40)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=gh,f^2=gh,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, 9620567);