#9214257

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

4542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9121453);