#9682777

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

4323 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=g,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, 9589973);