#9607824

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

4416 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)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2,5,4)(3,7,8,6)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(10,17)(11,21)(12,23)(19,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3,5,8)(2,7,4,6)(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,2)(3,6,8,7)(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=hi,e^2=g,f^2=ghi,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, 9515020);