#9730654

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	#51901	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,37)(34,40)(35,38)(36,39);(1,2,6,5)(3,7,4,8)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,22,24,17)(25,27,31,34)(26,30,32,40)(28,35,37,39)(29,38,33,36);(1,2,6,5)(3,8,4,7)(26,32)(27,34)(28,37)(36,38);(1,3,6,4)(2,7,5,8)(9,12,14,20)(10,16,18,24)(11,13,21,23)(15,22,19,17)(25,28)(26,33)(27,35)(29,32)(30,36)(31,37)(34,39)(38,40);(1,4,6,3)(2,7,5,8)(9,13)(10,17)(11,20)(12,21)(14,23)(15,16)(18,22)(19,24)(25,29,31,33)(26,28,32,37)(27,36,34,38)(30,39,40,35);(1,5,6,2)(3,8,4,7)(9,10)(11,15)(12,16)(13,22)(14,18)(17,23)(19,21)(20,24)(25,30)(26,34)(27,32)(28,36)(29,35)(31,40)(33,39)(37,38)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=h,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, 9637850);