#9523817

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

4345 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=g,e^2=ghi,f^2=g,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, 9431013);