#9078805

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8986001);