#10047437

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,30,33)(27,31,35,29)(28,32,36,39)(34,37,40,38);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,30,35)(26,29,33,31)(28,34,36,40)(32,38,39,37);(1,2,4,6)(3,5,8,7)(10,11)(15,17)(16,18)(22,23)(28,36)(32,39)(34,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,40)(38,39);(1,2)(3,7)(4,6)(5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,31)(26,27,33,35)(28,37,36,38)(32,40,39,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=i,f^2=gh,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, 9954633);