#10040811

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	#55228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=h,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, 9948007);