#9014323

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

4454 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)(3,8)(4,7)(5,6)(9,10,14,16)(11,19,20,15)(12,21,22,17)(13,18,23,24)(25,26,30,34)(27,29,33,31)(28,32,38,36)(35,37,40,39);(3,4)(7,8)(9,11,14,20)(10,15,16,19)(12,13,22,23)(17,18,21,24)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(1,3)(2,7)(4,6)(5,8)(10,16)(12,22)(13,23)(15,19)(25,26,30,34)(27,31,33,29)(28,36,38,32)(35,37,40,39);(3,4)(7,8)(9,12,14,22)(10,17,16,21)(11,13,20,23)(15,18,19,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,37)(39,40);(1,4,6,3)(2,7,5,8)(9,13)(10,18)(11,12)(14,23)(15,17)(16,24)(19,21)(20,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,5,6,2)(3,7,4,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,22)(23,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8921519);