#8418490

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

4443 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)(6,8)(9,10,14,17)(11,13,16,18)(12,15,23,21)(19,22,20,24)(25,26)(27,37)(28,30)(29,34)(31,35)(32,36)(33,40)(38,39);(3,7)(6,8)(9,11)(10,13)(12,20)(14,16)(15,24)(17,18)(19,23)(21,22)(25,27,32,40)(26,33,36,37)(28,38,29,35)(30,31,34,39);(1,2)(3,6)(4,5)(7,8)(11,16)(13,18)(19,20)(22,24)(25,28,32,29)(26,34,36,30)(27,38,40,35)(31,33,39,37);(1,3,4,7)(2,6,5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,23)(19,22)(20,24)(25,29,32,28)(26,34,36,30)(27,35,40,38)(31,33,39,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,21)(18,24)(25,30,32,34)(26,29,36,28)(27,39,40,31)(33,38,37,35);(9,13,14,18)(10,16,17,11)(12,22,23,24)(15,20,21,19)(25,31,32,39)(26,35,36,38)(27,34,40,30)(28,37,29,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=g,f^2=gi,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, 8325686);