#9039123

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	#40646	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946319);