#8288347

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26,31,32)(27,34,29,33)(28,37,38,30)(35,40,39,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(26,32)(28,38)(33,34)(35,39);(2,5)(6,8)(9,11,14,12)(10,16,18,15)(13,19,21,20)(17,23,24,22)(25,27,31,29)(26,33,32,34)(28,35,38,39)(30,40,37,36);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,28)(26,30)(27,35)(29,39)(31,38)(32,37)(33,40)(34,36);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,29,31,27)(26,34,32,33)(28,35,38,39)(30,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,38)(33,39)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,e^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, 8195543);