#8873718

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

4334 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780914);