#10040580

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9947776);