#9719237

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	#36909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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:

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

Polycyclic presentation:

2^9|c^2=i,e^2=ghi,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, 9626433);