#8132879

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

4356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=h,f^2=ghi,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, 8040075);