#8406710

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

4395 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^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, 8313906);