#9064867

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

4428 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,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, 8972063);