#9726614

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

4472 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)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,3,5,7)(2,4,8,6)(9,11,14,13)(10,15,16,18)(12,20,21,19)(17,24,23,22)(25,27)(26,32)(28,30)(29,38)(31,39)(33,40)(34,35)(36,37);(1,3)(2,6)(4,8)(5,7)(10,16)(15,18)(17,23)(22,24)(25,28,31,29)(26,33,36,34)(27,30,39,38)(32,40,37,35);(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,29,31,28)(26,34,36,33)(27,38,39,30)(32,35,37,40);(1,3)(2,4)(5,7)(6,8)(9,12,14,21)(10,17,16,23)(11,19,13,20)(15,22,18,24)(25,30,31,38)(26,35,36,40)(27,28,39,29)(32,34,37,33);(1,4)(2,7)(3,8)(5,6)(9,13,14,11)(10,18,16,15)(12,19,21,20)(17,22,23,24)(25,26)(27,32)(28,33)(29,34)(30,40)(31,36)(35,38)(37,39)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=hi,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, 9633810);