#9545594

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

4613 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452790);