#8276834

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

4485 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8184030);