#9123923

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4232 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=h,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, 9031119);