#10171757

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

4488 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=ghi,e^2=gi,f^2=g,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, 10078953);