#9206669

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

4531 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113865);