#9682769

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
prime radical	#40499	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4192 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=h,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, 9589965);