#10040819

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,6)(4,7)(5,8)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26,32,30)(27,33,38,40)(28,36,31,34)(29,39,37,35);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,27,32,38)(26,33,30,40)(28,29,31,37)(34,35,36,39);(1,3)(2,6)(4,8)(5,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,28,32,31)(26,34,30,36)(27,37,38,29)(33,39,40,35);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,29,32,37)(26,35,30,39)(27,28,38,31)(33,34,40,36);(1,2)(3,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,30,32,26)(27,33,38,40)(28,34,31,36)(29,39,37,35);(1,3)(2,6)(4,8)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,31,32,28)(26,36,30,34)(27,37,38,29)(33,39,40,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 9948015);