#9015467

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
prime radical	#54923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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)(3,8)(4,7)(5,6)(9,10)(11,12)(13,21)(14,16)(15,17)(18,22)(19,23)(20,24)(25,26,31,36)(27,35,30,32)(28,29,34,33)(37,40,39,38);(3,4)(7,8)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(25,27,31,30)(26,32,36,35)(28,39,34,37)(29,40,33,38);(1,3)(2,7)(4,6)(5,8)(10,16)(12,15)(13,22)(19,24)(25,28,31,34)(26,33,36,29)(27,37,30,39)(32,40,35,38);(3,4)(7,8)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39);(1,4,6,3)(2,7,5,8)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,22)(23,24)(25,29,31,33)(26,34,36,28)(27,38,30,40)(32,39,35,37);(1,5,6,2)(3,7,4,8)(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,30,31,27)(26,35,36,32)(28,39,34,37)(29,40,33,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,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, 8922663);