#9015298

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

4475 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=ghi,e^2=gi,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, 8922494);