#9729222

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	#52032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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)(5,8)(6,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,22)(20,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(10,15)(12,20)(16,22)(18,21)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(1,2,5,8)(3,4,7,6)(10,15)(11,19)(17,24)(18,21);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,16,15,22)(12,18,20,21)(13,17,23,24)(25,28,30,37)(26,32,33,39)(27,35,29,36)(31,38,34,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,22)(14,20)(15,24)(16,23)(19,21)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,39,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,18,15,21)(11,17,19,24)(12,16,20,22)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,40)(37,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=ghi,f^2=hi,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, 9636418);