#10038465

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

4677 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)(3,4)(5,6)(7,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,29)(28,34)(30,32)(31,33)(35,38)(36,39)(37,40);(1,3,6,7)(2,4,5,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,31,30)(26,32,33,29)(28,36,37,35)(34,38,40,39);(1,4,6,8)(2,7,5,3)(10,14)(15,18)(16,20)(22,24)(26,33)(28,37)(29,32)(35,36);(1,2,6,5)(3,8,7,4)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28,31,37)(26,34,33,40)(27,35,30,36)(29,38,32,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,29,31,32)(26,27,33,30)(28,38,37,39)(34,35,40,36);(1,5,6,2)(3,8,7,4)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,27)(26,29,33,32)(28,36,37,35)(34,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9945661);