#10173550

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

4705 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)(4,5)(6,8)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26,30,32)(27,34,37,31)(28,36,38,33)(29,39,40,35);(3,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,37)(26,31,32,34)(28,35,38,39)(29,36,40,33);(1,3)(2,6)(4,7)(5,8)(10,14)(15,17)(16,20)(22,23)(26,32)(28,38)(31,34)(35,39);(3,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,38)(26,33,32,36)(27,35,37,39)(29,34,40,31);(1,3)(2,7)(4,6)(5,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27)(26,34)(28,39)(29,33)(30,37)(31,32)(35,38)(36,40);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,35)(27,36)(28,34)(30,40)(31,38)(32,39)(33,37)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10080746);