#8936772

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

4553 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8843968);