#9161163

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	#48106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068359);