#10036740

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

4579 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)(27,35)(28,30)(29,33)(31,32)(34,40)(36,37)(38,39);(1,2)(3,8)(4,7)(5,6)(11,17)(15,21)(18,23)(22,24)(25,27,31,33)(26,29,32,35)(28,38,34,36)(30,37,40,39);(2,6)(3,7)(10,14)(15,21)(16,19)(22,24)(26,32)(28,34)(29,35)(36,38);(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,34)(26,30,32,40)(27,36,33,38)(29,39,35,37);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,35)(26,33,32,27)(28,39,34,37)(30,38,40,36);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,34)(27,37)(28,32)(29,36)(31,40)(33,39)(35,38)

Polycyclic presentation:

2^9|b^2=g,d^2=gi,e^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, 9943936);