#9250234

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

4238 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)(11,17)(12,22)(13,18)(14,19)(15,16)(20,21)(23,24)(26,31)(27,33)(29,37)(35,40);(1,2,5,7)(3,4,8,6)(10,16)(14,18)(17,22)(21,24)(25,26,30,31)(27,28,33,36)(29,38,37,32)(34,40,39,35);(1,3,5,8)(2,6,7,4)(9,11,15,12)(10,17,16,22)(13,23,19,20)(14,21,18,24)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(9,12,15,11)(10,17,16,22)(13,23,19,20)(14,24,18,21)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,19)(10,18,16,14)(11,20,12,23)(17,21,22,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,14,15,18)(10,19,16,13)(11,21,12,24)(17,20,22,23)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9157430);