#8126097

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

4890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8033293);