#9607971

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

4616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9515167);