#9435068

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	#38922	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22)(26,31)(27,35)(28,37)(34,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,21,24,23)(25,26)(27,32)(28,33)(29,39)(30,31)(34,38)(35,36)(37,40);(2,5)(6,8)(10,17)(11,12)(14,24)(20,22)(25,27,30,35)(26,32,31,36)(28,38,37,29)(33,34,40,39);(3,7)(6,8)(9,12,15,11)(10,18,17,16)(13,20,19,22)(14,23,24,21)(25,28,30,37)(26,33,31,40)(27,29,35,38)(32,39,36,34);(1,3)(2,6)(4,7)(5,8)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,23,18,21)(25,29)(26,34)(27,28)(30,38)(31,39)(32,40)(33,36)(35,37);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,20)(18,22)(25,26)(27,36)(28,33)(29,34)(30,31)(32,35)(37,40)(38,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9342264);