#9017442

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

4575 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,31)(27,34)(33,39)(35,37);(3,8)(6,7)(10,14)(15,17)(16,20)(22,23)(25,26,30,31)(27,32,34,29)(28,33,36,39)(35,40,37,38);(1,3,5,8)(2,6,4,7)(10,14)(11,18)(12,21)(22,23)(27,34)(28,36)(29,32)(33,39);(3,8)(6,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,32)(28,37)(29,31)(30,34)(33,38)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28,30,36)(26,33,31,39)(27,35,34,37)(29,38,32,40);(1,4,5,2)(3,6,8,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,32)(26,27,31,34)(28,38,36,40)(33,35,39,37)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924638);