#10040767

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

4017 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,24)(22,23)(25,26,31,29)(27,34,30,32)(28,33,38,37)(35,40,36,39);(1,3)(2,5)(4,6)(7,8)(9,11,15,21)(10,16,19,14)(12,20,13,23)(17,22,18,24)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(2,7)(3,4)(9,12,15,13)(10,17,19,18)(11,20,21,23)(14,24,16,22)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(1,3,6,4)(2,8,7,5)(9,13,15,12)(10,18,19,17)(11,20,21,23)(14,24,16,22)(25,28,31,38)(26,33,29,37)(27,35,30,36)(32,39,34,40);(1,4,6,3)(2,8,7,5)(9,14)(10,11)(12,22)(13,24)(15,16)(17,23)(18,20)(19,21)(25,29,31,26)(27,32,30,34)(28,37,38,33)(35,39,36,40);(1,5,6,8)(2,4,7,3)(9,10)(11,16)(12,17)(13,18)(14,21)(15,19)(20,22)(23,24)(25,30,31,27)(26,34,29,32)(28,36,38,35)(33,40,37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gi,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, 9947963);