#9015347

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=ghi,e^2=i,f^2=h,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, 8922543);