#9243746

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

4767 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)(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,15)(16,19)(18,21)(20,22)(23,24)(25,26,31,29)(27,32,37,39)(28,34,38,33)(30,40,36,35);(1,3)(2,8)(4,6)(5,7)(10,15)(13,16)(17,22)(21,23)(25,27,31,37)(26,32,29,39)(28,30,38,36)(33,35,34,40);(1,2,7,6)(3,8,5,4)(12,20)(17,22)(18,24)(21,23)(25,28,31,38)(26,33,29,34)(27,36,37,30)(32,40,39,35);(1,4)(2,5)(3,6)(7,8)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,28)(26,34)(27,30)(29,33)(31,38)(32,40)(35,39)(36,37);(1,5,7,3)(2,8,6,4)(9,12)(10,17)(11,18)(13,21)(14,20)(15,22)(16,23)(19,24)(25,29,31,26)(27,32,37,39)(28,33,38,34)(30,40,36,35);(1,6,7,2)(3,8,5,4)(9,13,14,16)(10,11,15,19)(12,21,20,23)(17,18,22,24)(25,30)(26,35)(27,28)(29,40)(31,36)(32,33)(34,39)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=h,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, 9150942);