#9683663

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

4656 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)(11,14)(12,17)(13,18)(15,16)(19,23)(20,22)(21,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,13)(15,19)(16,23)(17,18)(20,21)(22,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,3,5,6)(2,4,7,8)(10,16)(11,19)(12,20)(18,24)(26,28)(31,33)(32,36)(39,40);(9,12,15,20)(10,17,16,22)(11,13,19,21)(14,18,23,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,18)(11,20)(12,19)(14,22)(15,21)(16,24)(17,23)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,23)(10,19,16,11)(12,18,20,24)(13,17,21,22)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39)

Polycyclic presentation:

2^9|c^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9590859);