#8159338

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

4788 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=g,f^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^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, 8066534);