#9789212

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	#36901	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696408);