#10464558

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

4416 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10371754);