#9015519

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=ghi,e^2=gi,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, 8922715);