#9713736

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	#38831	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=gi,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, 9620932);