#9214763

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

4645 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9121959);