#9016457

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	#41007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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)(3,5)(4,8)(6,7)(9,10)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,37,39,40);(2,7)(3,8)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,3,6,8)(2,4,7,5)(10,14)(12,21)(13,20)(16,19)(25,27,30,38)(26,32,35,31)(28,39,34,36)(29,40,33,37);(1,4,6,5)(2,3,7,8)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,2)(3,4)(5,8)(6,7)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28,30,34)(26,33,35,29)(27,36,38,39)(31,37,32,40);(1,5,6,4)(2,8,7,3)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8923653);