#9788653

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9695849);