#8274168

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	#37725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4399 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8181364);