#9716822

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
prime radical	#37829	C₂
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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^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, 9624018);