#9016534

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

4480 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8923730);