#8521628

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

4640 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,16,19,13)(12,14,17,23)(20,21,24,22)(26,30)(27,33)(32,35)(34,39);(2,5)(6,7)(9,11,15,19)(10,16,18,13)(12,20,17,24)(14,21,23,22)(25,26,29,30)(27,31,33,37)(28,32,36,35)(34,38,39,40);(1,2)(3,6)(4,5)(7,8)(11,19)(13,16)(20,24)(21,22)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,20)(13,22)(15,17)(16,21)(18,23)(19,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,13)(10,11)(12,22)(14,20)(15,16)(17,21)(18,19)(23,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(9,14)(10,17)(11,21)(12,18)(13,20)(15,23)(16,24)(19,22)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8428824);