#9015507

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

4266 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8922703);