#10036776

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

4494 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)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,29)(28,34)(30,33)(31,35)(32,36)(37,39)(38,40);(1,2)(3,8)(4,7)(5,6)(11,17)(15,21)(18,23)(22,24)(25,27,32,30)(26,33,36,29)(28,37,31,38)(34,40,35,39);(2,6)(3,7)(10,14)(15,21)(16,19)(22,24)(25,28,32,31)(26,34,36,35)(27,37,30,38)(29,39,33,40);(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,32,33)(26,27,36,30)(28,39,31,40)(34,37,35,38);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,30,32,27)(26,33,36,29)(28,37,31,38)(34,39,35,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,31,32,28)(26,35,36,34)(27,38,30,37)(29,40,33,39)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^2=g,f^2=g,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, 9943972);