#8406756

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

4621 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313952);