#9607217

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9514413);