#8873683

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	#50530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780879);