#8743416

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

4196 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8650612);