#8760367

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	#50744	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4345 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)(13,19)(14,24)(15,17)(20,23)(21,22)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,40)(36,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,17,18)(13,22,24,20)(14,23,19,21)(26,31)(28,33)(32,34)(35,39);(2,5)(6,7)(10,17)(11,12)(13,24)(21,23)(25,26,30,31)(27,32,37,34)(28,29,33,38)(35,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,18,17,16)(13,22,24,20)(14,23,19,21)(25,27,30,37)(26,32,31,34)(28,35,33,39)(29,36,38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,23)(18,21)(25,28,30,33)(26,29,31,38)(27,35,37,39)(32,36,34,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,20,18,22)(25,29)(26,33)(27,36)(28,31)(30,38)(32,39)(34,35)(37,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8667563);