#9064871

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	#35923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4348 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)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(1,2)(3,4)(5,6)(7,8)(9,11,15,18)(10,16,17,12)(13,22,19,20)(14,21,24,23)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(2,6)(3,7)(10,17)(11,18)(13,19)(21,23)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(3,7)(4,8)(9,12,15,16)(10,18,17,11)(13,23,19,21)(14,20,24,22)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,19)(10,14,17,24)(11,20,18,22)(12,21,16,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,4)(2,3)(5,8)(6,7)(9,14)(10,19)(11,21)(12,22)(13,17)(15,24)(16,20)(18,23)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972067);