#8132885

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

4360 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)(12,14)(13,22)(15,17)(18,24)(19,23)(20,21)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,17,22)(12,21,18,19)(14,20,24,23)(26,27)(31,33)(32,34)(38,39);(2,5)(6,8)(10,17)(11,13)(12,18)(20,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,18)(10,14,17,24)(11,19,13,21)(16,23,22,20)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,16,17,22)(12,21,18,19)(14,23,24,20)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,3,4,7)(2,6,5,8)(9,14)(10,18)(11,20)(12,17)(13,23)(15,24)(16,19)(21,22)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=gh,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, 8040081);