#9029691

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

4567 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)(3,4)(5,6)(7,8)(9,10,15,13)(11,12,19,16)(14,21,22,17)(18,23,24,20)(25,26,31,34)(27,35,37,32)(28,38,39,29)(30,33,36,40);(2,6)(4,7)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,27,31,37)(26,32,34,35)(28,33,39,40)(29,36,38,30);(1,2,5,6)(3,7,8,4)(10,13)(12,16)(14,22)(18,24)(28,39)(29,38)(30,36)(33,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,28,31,39)(26,29,34,38)(27,33,37,40)(30,32,36,35);(1,4,5,7)(2,3,6,8)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,29)(26,28)(27,36)(30,37)(31,38)(32,33)(34,39)(35,40);(1,3)(2,7)(4,6)(5,8)(9,14,15,22)(10,17,13,21)(11,18,19,24)(12,20,16,23)(25,30,31,36)(26,33,34,40)(27,29,37,38)(28,32,39,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=gh,e^2=h,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, 8936887);