#9040405

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

4444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947601);