#9230865

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
prime radical	#51899	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4594 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=gh,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, 9138061);