#9039268

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	#52013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gi,e^2=g,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, 8946464);