#9028794

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

4261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=h,e^2=gh,f^2=gi,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, 8935990);