#9607534

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

4756 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)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26)(27,30)(28,32)(29,31)(33,34)(35,38)(36,37)(39,40);(1,2,5,4)(3,7,8,6)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,40)(37,39);(1,3,5,8)(2,6,4,7)(10,14)(12,21)(13,20)(16,19)(26,31)(27,34)(28,36)(38,39);(1,3,5,8)(2,7,4,6)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,28,29,36)(26,32,31,37)(27,35,34,40)(30,38,33,39);(1,3)(2,7)(4,6)(5,8)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,26)(27,33)(28,37)(29,31)(30,34)(32,36)(35,38)(39,40);(1,4,5,2)(3,6,8,7)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,27)(26,33)(28,35)(29,34)(30,31)(32,39)(36,40)(37,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,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, 9514730);