#8509781

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

4398 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=g,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, 8416977);