#9105322

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

4311 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012518);