#8406694

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

4598 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)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(3,7)(6,8)(10,12)(11,17)(16,20)(18,23)(25,27,31,36)(26,32,35,40)(28,39,33,30)(29,34,38,37);(1,2)(3,6)(4,5)(7,8)(13,21)(16,20)(18,23)(22,24)(27,36)(30,39)(32,40)(34,37);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,33)(32,37)(34,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,39)(34,36);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,34)(27,28)(29,40)(31,39)(32,38)(33,36)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,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, 8313890);