#9713472

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

4271 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,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, 9620668);