#9730642

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

4291 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=g,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, 9637838);