#9726623

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

4306 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,34)(28,30)(29,33)(31,35)(32,39)(36,40)(37,38);(2,5)(6,7)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,24,20,22)(25,27)(26,32)(28,36)(29,37)(30,38)(31,39)(33,40)(34,35);(1,2)(3,6)(4,5)(7,8)(10,12)(11,18)(17,20)(19,21)(25,28,31,29)(26,30,35,33)(27,36,39,37)(32,38,34,40);(1,3,4,8)(2,6,5,7)(9,12,15,10)(11,14,18,16)(13,20,23,17)(19,24,21,22)(25,29,31,28)(26,33,35,30)(27,37,39,36)(32,40,34,38);(9,13,15,23)(10,17,12,20)(11,19,18,21)(14,24,16,22)(25,27)(26,34)(28,36)(29,37)(30,40)(31,39)(32,35)(33,38);(9,14)(10,11)(12,18)(13,22)(15,16)(17,21)(19,20)(23,24)(25,30,31,33)(26,28,35,29)(27,38,39,40)(32,36,34,37)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633819);