#8159192

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

4456 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)(3,4)(9,10)(11,14)(12,17)(13,18)(15,20)(16,21)(19,22)(23,24)(25,26,30,32)(27,31,33,28)(29,37,38,34)(35,36,40,39);(1,2,6,5)(3,8,4,7)(10,11)(15,17)(16,18)(22,23)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,39,37,36);(1,3,6,4)(2,7,5,8)(10,11)(15,17)(16,18)(22,23)(26,32)(27,33)(34,37)(35,40);(1,4,6,3)(2,8,5,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,31)(28,32)(29,35)(30,33)(34,39)(36,37)(38,40);(1,5,6,2)(3,8,4,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,40,37,35);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,38)(26,34,32,37)(27,35,33,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=gh,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, 8066388);