#8421702

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

4659 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8328898);