#8301424

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	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,e^2=i,f^2=g,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, 8208620);