#9683766

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

4339 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)(11,19)(12,15)(13,17)(14,22)(16,20)(18,23)(21,24)(26,31)(32,33)(34,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,22,21,18)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(10,12)(16,19)(17,21)(18,22)(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(9,12,15,10)(11,19,20,16)(13,21,24,17)(14,22,23,18)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2)(3,4)(5,7)(6,8)(9,13,15,24)(10,17,12,21)(11,14,20,23)(16,18,19,22)(25,28,30,27)(26,33,31,32)(29,35,37,36)(34,39,38,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,13)(12,22)(15,23)(16,17)(19,21)(20,24)(25,29,30,37)(26,34,31,38)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|b^2=h,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, 9590962);