#10521370

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428566);