#9787193

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

4738 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)(11,17)(12,13)(14,15)(16,20)(18,23)(19,24)(21,22)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,37)(38,39);(1,2,4,6)(3,5,8,7)(10,15)(12,21)(16,17)(18,24)(25,27)(26,29)(28,36)(30,34)(31,33)(32,40)(35,38)(37,39);(1,2)(3,5)(4,6)(7,8)(26,31)(27,34)(32,39)(35,36);(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(1,2)(3,7)(4,6)(5,8)(9,12,14,21)(10,13,15,22)(11,18,20,24)(16,19,17,23)(25,28,30,38)(26,32,31,39)(27,35,34,36)(29,37,33,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,39)(38,40)

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9694389);