#10352531

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

4372 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10259727);