#9446296

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

4498 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9353492);